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      <title>【解析几何】准线与焦点</title>
      <link href="/2024/08/06/%E3%80%90%E8%A7%A3%E6%9E%90%E5%87%A0%E4%BD%95%E3%80%91%E5%87%86%E7%BA%BF%E4%B8%8E%E7%84%A6%E7%82%B9/"/>
      <url>/2024/08/06/%E3%80%90%E8%A7%A3%E6%9E%90%E5%87%A0%E4%BD%95%E3%80%91%E5%87%86%E7%BA%BF%E4%B8%8E%E7%84%A6%E7%82%B9/</url>
      
        <content type="html"><![CDATA[<h1 id="解析几何准线与焦点"><a class="markdownIt-Anchor" href="#解析几何准线与焦点"></a> 【解析几何】准线与焦点</h1><p>在学习圆锥曲线的过程中，我们必然会遇到诸多有关焦点这个特殊点的题目。比如下面这道：</p><p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mtext>设</mtext><msub><mi>F</mi><mn>1</mn></msub><mo separator="true">,</mo><msub><mi>F</mi><mn>2</mn></msub><mtext>分别为椭圆</mtext><mfrac><msup><mi>x</mi><mn>2</mn></msup><mn>3</mn></mfrac><mo>+</mo><msup><mi>y</mi><mn>2</mn></msup><mo>=</mo><mn>1</mn><mtext>的左、右焦点，点</mtext><mi>A</mi><mo separator="true">,</mo><mi>B</mi><mspace linebreak="newline"></mspace><mtext>在椭圆上，若</mtext><mover accent="true"><mrow><msub><mi>F</mi><mn>1</mn></msub><mi>A</mi></mrow><mo stretchy="true">→</mo></mover><mo>=</mo><mn>5</mn><mover accent="true"><mrow><msub><mi>F</mi><mn>2</mn></msub><mi>B</mi></mrow><mo stretchy="true">→</mo></mover><mo separator="true">,</mo><mtext>则点A的坐标是</mtext><mo stretchy="false">(</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\text{设}F_1,F_2\text{分别为椭圆}\frac{x^2}{3}+y^2=1\text{的左、右焦点，点}A,B\\\text{在椭圆上，若}\overrightarrow{F_1A}=5\overrightarrow{F_2B},\text{则点A的坐标是}()</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.177108em;vertical-align:-0.686em;"></span><span class="mord text"><span class="mord cjk_fallback">设</span></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord text"><span class="mord cjk_fallback">分别为椭圆</span></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.491108em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">3</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1.0585479999999998em;vertical-align:-0.19444em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641079999999999em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.8777699999999999em;vertical-align:-0.19444em;"></span><span class="mord">1</span><span class="mord text"><span class="mord cjk_fallback">的左、右焦点，点</span></span><span class="mord mathnormal">A</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span><span class="mspace newline"></span><span class="base"><span class="strut" style="height:1.35533em;vertical-align:-0.15em;"></span><span class="mord text"><span class="mord cjk_fallback">在椭圆上，若</span></span><span class="mord accent"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.20533em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord mathnormal">A</span></span></span><span class="svg-align" style="top:-3.6833299999999998em;"><span class="pstrut" style="height:3em;"></span><span class="hide-tail" style="height:0.522em;min-width:0.888em;"><svg width='400em' height='0.522em' viewBox='0 0 400000 522' preserveAspectRatio='xMaxYMin slice'><path d='M0 241v40h399891c-47.3 35.3-84 78-110 128-16.7 32-27.7 63.7-33 95 0 1.3-.2 2.7-.5 4-.3 1.3-.5 2.3-.5 3 0 7.3 6.7 11 20 11 8 0 13.2-.8 15.5-2.5 2.3-1.7 4.2-5.5 5.5-11.5 2-13.3 5.7-27 11-41 14.7-44.7 39-84.5 73-119.5s73.7-60.2 119-75.5c6-2 9-5.7 9-11s-3-9-9-11c-45.3-15.3-85-40.5-119-75.5s-58.3-74.8-73-119.5c-4.7-14-8.3-27.3-11-40-1.3-6.7-3.2-10.8-5.5-12.5-2.3-1.7-7.5-2.5-15.5-2.5-14 0-21 3.7-21 11 0 2 2 10.3 6 25 20.7 83.3 67 151.7 139 205zm0 0v40h399900v-40z'/></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1.45533em;vertical-align:-0.25em;"></span><span class="mord">5</span><span class="mord accent"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.20533em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span><span class="svg-align" style="top:-3.6833299999999998em;"><span class="pstrut" style="height:3em;"></span><span class="hide-tail" style="height:0.522em;min-width:0.888em;"><svg width='400em' height='0.522em' viewBox='0 0 400000 522' preserveAspectRatio='xMaxYMin slice'><path d='M0 241v40h399891c-47.3 35.3-84 78-110 128-16.7 32-27.7 63.7-33 95 0 1.3-.2 2.7-.5 4-.3 1.3-.5 2.3-.5 3 0 7.3 6.7 11 20 11 8 0 13.2-.8 15.5-2.5 2.3-1.7 4.2-5.5 5.5-11.5 2-13.3 5.7-27 11-41 14.7-44.7 39-84.5 73-119.5s73.7-60.2 119-75.5c6-2 9-5.7 9-11s-3-9-9-11c-45.3-15.3-85-40.5-119-75.5s-58.3-74.8-73-119.5c-4.7-14-8.3-27.3-11-40-1.3-6.7-3.2-10.8-5.5-12.5-2.3-1.7-7.5-2.5-15.5-2.5-14 0-21 3.7-21 11 0 2 2 10.3 6 25 20.7 83.3 67 151.7 139 205zm0 0v40h399900v-40z'/></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord text"><span class="mord cjk_fallback">则点</span><span class="mord">A</span><span class="mord cjk_fallback">的坐标是</span></span><span class="mopen">(</span><span class="mclose">)</span></span></span></span></span></p><p><img src="https://cdn.jsdelivr.net/gh/Yurchiu/PicGo/90e0c08c397e28f619fa0dcbc8bad280.png" alt="" /></p><p>看到这里，很多同学可能就会选择联立求解，如下。</p><p><strong>解法一：</strong></p><p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mtext>由于对称性将</mtext><mover accent="true"><mrow><msub><mi>F</mi><mn>2</mn></msub><mi>B</mi></mrow><mo stretchy="true">→</mo></mover><mtext>平移到左焦点，设</mtext><mi>A</mi><mo stretchy="false">(</mo><msub><mi>x</mi><mn>1</mn></msub><mo separator="true">,</mo><msub><mi>y</mi><mn>1</mn></msub><mo stretchy="false">)</mo><mo separator="true">,</mo><mi>B</mi><mo stretchy="false">(</mo><msub><mi>x</mi><mn>2</mn></msub><mo separator="true">,</mo><msub><mi>y</mi><mn>2</mn></msub><mo stretchy="false">)</mo><mspace linebreak="newline"></mspace><mtext>由于直线</mtext><mi>A</mi><msub><mi>B</mi><mn>1</mn></msub><mtext>过左焦点，故设直线</mtext><mi>A</mi><msub><mi>B</mi><mn>1</mn></msub><mo>:</mo><mi>x</mi><mo>=</mo><mi>m</mi><mi>y</mi><mo>−</mo><msqrt><mn>2</mn></msqrt><mi mathvariant="normal">.</mi><mspace linebreak="newline"></mspace><mtext>联立</mtext><mrow><mo fence="true">{</mo><mtable rowspacing="0.3599999999999999em" columnalign="left left" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mi>x</mi><mo>=</mo><mi>m</mi><mi>y</mi><mo>−</mo><msqrt><mn>2</mn></msqrt></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>3</mn><msup><mi>y</mi><mn>2</mn></msup><mo>=</mo><mn>3</mn></mrow></mstyle></mtd></mtr></mtable></mrow><mo>⟹</mo><mo stretchy="false">(</mo><mi>m</mi><mi>y</mi><mo>−</mo><msqrt><mn>2</mn></msqrt><msup><mo stretchy="false">)</mo><mn>2</mn></msup><mo>+</mo><mn>3</mn><msup><mi>y</mi><mn>2</mn></msup><mo>=</mo><mn>3</mn></mrow><annotation encoding="application/x-tex">\text{由于对称性将}\overrightarrow{F_2B}\text{平移到左焦点，设}A(x_1,y_1),B(x_2,y_2)\\\text{由于直线}AB_1\text{过左焦点，故设}\text{直线}AB_1:x=my-\sqrt{2}.\\\text{联立} \begin{cases} x=my-\sqrt{2}\\ x^2+3y^2=3 \end{cases}\Longrightarrow(my-\sqrt{2})^2+3y^2=3</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.45533em;vertical-align:-0.25em;"></span><span class="mord text"><span class="mord cjk_fallback">由于对称性将</span></span><span class="mord accent"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.20533em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span><span class="svg-align" style="top:-3.6833299999999998em;"><span class="pstrut" style="height:3em;"></span><span class="hide-tail" style="height:0.522em;min-width:0.888em;"><svg width='400em' height='0.522em' viewBox='0 0 400000 522' preserveAspectRatio='xMaxYMin slice'><path d='M0 241v40h399891c-47.3 35.3-84 78-110 128-16.7 32-27.7 63.7-33 95 0 1.3-.2 2.7-.5 4-.3 1.3-.5 2.3-.5 3 0 7.3 6.7 11 20 11 8 0 13.2-.8 15.5-2.5 2.3-1.7 4.2-5.5 5.5-11.5 2-13.3 5.7-27 11-41 14.7-44.7 39-84.5 73-119.5s73.7-60.2 119-75.5c6-2 9-5.7 9-11s-3-9-9-11c-45.3-15.3-85-40.5-119-75.5s-58.3-74.8-73-119.5c-4.7-14-8.3-27.3-11-40-1.3-6.7-3.2-10.8-5.5-12.5-2.3-1.7-7.5-2.5-15.5-2.5-14 0-21 3.7-21 11 0 2 2 10.3 6 25 20.7 83.3 67 151.7 139 205zm0 0v40h399900v-40z'/></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span><span class="mord text"><span class="mord cjk_fallback">平移到左焦点，设</span></span><span class="mord mathnormal">A</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span><span class="mspace newline"></span><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em;"></span><span class="mord text"><span class="mord cjk_fallback">由于直线</span></span><span class="mord mathnormal">A</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.05017em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord text"><span class="mord cjk_fallback">过左焦点，故设</span></span><span class="mord text"><span class="mord cjk_fallback">直线</span></span><span class="mord mathnormal">A</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.05017em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">:</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.7777700000000001em;vertical-align:-0.19444em;"></span><span class="mord mathnormal">m</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1.04em;vertical-align:-0.08390500000000001em;"></span><span class="mord sqrt"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.956095em;"><span class="svg-align" style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord" style="padding-left:0.833em;"><span class="mord">2</span></span></span><span style="top:-2.916095em;"><span class="pstrut" style="height:3em;"></span><span class="hide-tail" style="min-width:0.853em;height:1.08em;"><svg width='400em' height='1.08em' viewBox='0 0 400000 1080' preserveAspectRatio='xMinYMin slice'><path d='M95,702c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z'/></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.08390500000000001em;"><span></span></span></span></span></span><span class="mord">.</span></span><span class="mspace newline"></span><span class="base"><span class="strut" style="height:3.0000299999999998em;vertical-align:-1.25003em;"></span><span class="mord text"><span class="mord cjk_fallback">联立</span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size4">{</span></span><span class="mord"><span class="mtable"><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.69em;"><span style="top:-3.69em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord mathnormal">m</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord sqrt"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.90722em;"><span class="svg-align" style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord" style="padding-left:0.833em;"><span class="mord">2</span></span></span><span style="top:-2.86722em;"><span class="pstrut" style="height:3em;"></span><span class="hide-tail" style="min-width:0.853em;height:1.08em;"><svg width='400em' height='1.08em' viewBox='0 0 400000 1080' preserveAspectRatio='xMinYMin slice'><path d='M95,702c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z'/></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.13278em;"><span></span></span></span></span></span></span></span><span style="top:-2.25em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord">3</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord">3</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.19em;"><span></span></span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">⟹</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathnormal">m</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1.206095em;vertical-align:-0.25em;"></span><span class="mord sqrt"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.956095em;"><span class="svg-align" style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord" style="padding-left:0.833em;"><span class="mord">2</span></span></span><span style="top:-2.916095em;"><span class="pstrut" style="height:3em;"></span><span class="hide-tail" style="min-width:0.853em;height:1.08em;"><svg width='400em' height='1.08em' viewBox='0 0 400000 1080' preserveAspectRatio='xMinYMin slice'><path d='M95,702c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z'/></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.08390500000000001em;"><span></span></span></span></span></span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641079999999999em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1.0585479999999998em;vertical-align:-0.19444em;"></span><span class="mord">3</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641079999999999em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">3</span></span></span></span></span></p><p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mtext>则</mtext><mrow><mo fence="true">{</mo><mtable rowspacing="0.3599999999999999em" columnalign="left left" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><msub><mi>y</mi><mn>1</mn></msub><mo>=</mo><mo>−</mo><mn>5</mn><msub><mi>y</mi><mn>2</mn></msub></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><msub><mi>y</mi><mn>1</mn></msub><mo>+</mo><msub><mi>y</mi><mn>2</mn></msub><mo>=</mo><mo>−</mo><mfrac><mrow><mn>2</mn><msqrt><mn>2</mn></msqrt><mi>m</mi></mrow><mrow><msup><mi>m</mi><mn>2</mn></msup><mo>+</mo><mn>3</mn></mrow></mfrac></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><msub><mi>y</mi><mn>1</mn></msub><msub><mi>y</mi><mn>2</mn></msub><mo>=</mo><mo>−</mo><mfrac><mn>1</mn><mrow><msup><mi>m</mi><mn>2</mn></msup><mo>+</mo><mn>1</mn></mrow></mfrac></mrow></mstyle></mtd></mtr></mtable></mrow><mo>⇒</mo><mrow><mo fence="true">{</mo><mtable rowspacing="0.15999999999999992em" columnalign="center" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><msub><mi>y</mi><mn>1</mn></msub><mo>=</mo><mo>−</mo><mfrac><mi>m</mi><mrow><msqrt><mn>2</mn></msqrt><mo stretchy="false">(</mo><msup><mi>m</mi><mn>2</mn></msup><mo>+</mo><mn>3</mn><mo stretchy="false">)</mo></mrow></mfrac></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><msub><mi>y</mi><mn>2</mn></msub><mo>=</mo><mfrac><mrow><mn>5</mn><mi>m</mi></mrow><mrow><msqrt><mn>2</mn></msqrt><mo stretchy="false">(</mo><msup><mi>m</mi><mn>2</mn></msup><mo>+</mo><mn>3</mn><mo stretchy="false">)</mo></mrow></mfrac></mrow></mstyle></mtd></mtr></mtable></mrow><mo>⇒</mo><mfrac><mrow><mn>5</mn><msup><mi>m</mi><mn>2</mn></msup></mrow><mrow><mn>2</mn><mo stretchy="false">(</mo><msup><mi>m</mi><mn>2</mn></msup><mo>+</mo><mn>3</mn><mo stretchy="false">)</mo></mrow></mfrac><mo>=</mo><mn>1</mn><mo>⇒</mo><mi>m</mi><mo>=</mo><mo>±</mo><msqrt><mn>2</mn></msqrt><mspace linebreak="newline"></mspace><mtext>所以点</mtext><mi>A</mi><mo stretchy="false">(</mo><mn>0</mn><mo separator="true">,</mo><mo>±</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\text{则}\begin{cases}y_1=-5y_2\\y_1+y_2=-\frac{2\sqrt{2}m}{m^2+3}\\y_1y_2=-\frac{1}{m^2+1}\end{cases}\Rightarrow\left\{     \begin{array}{c}       y_1=-\frac{m}{\sqrt{2}(m^2+3)} \\        y_2=\frac{5m}{\sqrt{2}(m^2+3)}\\     \end{array}\right.\Rightarrow\frac{5m^2}{2(m^2+3)}=1\Rightarrowm=\pm\sqrt{2}\\\text{所以点}A(0,\pm1)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:4.35em;vertical-align:-1.9249999999999998em;"></span><span class="mord text"><span class="mord cjk_fallback">则</span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="minner"><span class="mopen"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.35002em;"><span style="top:-2.19999em;"><span class="pstrut" style="height:3.15em;"></span><span class="delimsizinginner delim-size4"><span>⎩</span></span></span><span style="top:-2.19499em;"><span class="pstrut" style="height:3.15em;"></span><span class="delimsizinginner delim-size4"><span>⎪</span></span></span><span style="top:-2.20499em;"><span class="pstrut" style="height:3.15em;"></span><span class="delimsizinginner delim-size4"><span>⎪</span></span></span><span style="top:-3.15001em;"><span class="pstrut" style="height:3.15em;"></span><span class="delimsizinginner delim-size4"><span>⎨</span></span></span><span style="top:-4.2950099999999996em;"><span class="pstrut" style="height:3.15em;"></span><span class="delimsizinginner delim-size4"><span>⎪</span></span></span><span style="top:-4.30501em;"><span class="pstrut" style="height:3.15em;"></span><span class="delimsizinginner delim-size4"><span>⎪</span></span></span><span style="top:-4.60002em;"><span class="pstrut" style="height:3.15em;"></span><span class="delimsizinginner delim-size4"><span>⎧</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.8500199999999998em;"><span></span></span></span></span></span></span><span class="mord"><span class="mtable"><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.425em;"><span style="top:-4.455em;"><span class="pstrut" style="height:3.038em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord">−</span><span class="mord">5</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span><span style="top:-2.985em;"><span class="pstrut" style="height:3.038em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord">−</span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0379999999999998em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathnormal mtight">m</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7463142857142857em;"><span style="top:-2.786em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mbin mtight">+</span><span class="mord mtight">3</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.3990085em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span><span class="mord sqrt mtight"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.912845em;"><span class="svg-align" style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mtight" style="padding-left:0.833em;"><span class="mord mtight">2</span></span></span><span style="top:-2.872845em;"><span class="pstrut" style="height:3em;"></span><span class="hide-tail mtight" style="min-width:0.853em;height:1.08em;"><svg width='400em' height='1.08em' viewBox='0 0 400000 1080' preserveAspectRatio='xMinYMin slice'><path d='M95,702c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z'/></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.12715500000000002em;"><span></span></span></span></span></span><span class="mord mathnormal mtight">m</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.403331em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span><span style="top:-1.545em;"><span class="pstrut" style="height:3.038em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord">−</span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.845108em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathnormal mtight">m</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7463142857142857em;"><span style="top:-2.786em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.403331em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.9249999999999998em;"><span></span></span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">⇒</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:3.0000299999999998em;vertical-align:-1.25003em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size4">{</span></span><span class="mord"><span class="mtable"><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.7165455em;"><span style="top:-3.8765454999999998em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord">−</span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.695392em;"><span style="top:-2.5510085em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord sqrt mtight"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.912845em;"><span class="svg-align" style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mtight" style="padding-left:0.833em;"><span class="mord mtight">2</span></span></span><span style="top:-2.872845em;"><span class="pstrut" style="height:3em;"></span><span class="hide-tail mtight" style="min-width:0.853em;height:1.08em;"><svg width='400em' height='1.08em' viewBox='0 0 400000 1080' preserveAspectRatio='xMinYMin slice'><path d='M95,702c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z'/></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.12715500000000002em;"><span></span></span></span></span></span><span class="mopen mtight">(</span><span class="mord mtight"><span class="mord mathnormal mtight">m</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7463142857142857em;"><span style="top:-2.786em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mbin mtight">+</span><span class="mord mtight">3</span><span class="mclose mtight">)</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">m</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.6239914999999999em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span><span style="top:-2.4074459999999998em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.845108em;"><span style="top:-2.5510085em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord sqrt mtight"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.912845em;"><span class="svg-align" style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mtight" style="padding-left:0.833em;"><span class="mord mtight">2</span></span></span><span style="top:-2.872845em;"><span class="pstrut" style="height:3em;"></span><span class="hide-tail mtight" style="min-width:0.853em;height:1.08em;"><svg width='400em' height='1.08em' viewBox='0 0 400000 1080' preserveAspectRatio='xMinYMin slice'><path d='M95,702c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z'/></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.12715500000000002em;"><span></span></span></span></span></span><span class="mopen mtight">(</span><span class="mord mtight"><span class="mord mathnormal mtight">m</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7463142857142857em;"><span style="top:-2.786em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mbin mtight">+</span><span class="mord mtight">3</span><span class="mclose mtight">)</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">5</span><span class="mord mathnormal mtight">m</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.6239914999999999em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.2165455em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">⇒</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:2.427108em;vertical-align:-0.936em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.491108em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">m</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.740108em;"><span style="top:-2.9890000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord">3</span><span class="mclose">)</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">5</span><span class="mord"><span class="mord mathnormal">m</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.936em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">1</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">⇒</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathnormal">m</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1.04em;vertical-align:-0.08390500000000001em;"></span><span class="mord">±</span><span class="mord sqrt"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.956095em;"><span class="svg-align" style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord" style="padding-left:0.833em;"><span class="mord">2</span></span></span><span style="top:-2.916095em;"><span class="pstrut" style="height:3em;"></span><span class="hide-tail" style="min-width:0.853em;height:1.08em;"><svg width='400em' height='1.08em' viewBox='0 0 400000 1080' preserveAspectRatio='xMinYMin slice'><path d='M95,702c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z'/></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.08390500000000001em;"><span></span></span></span></span></span></span><span class="mspace newline"></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord text"><span class="mord cjk_fallback">所以点</span></span><span class="mord mathnormal">A</span><span class="mopen">(</span><span class="mord">0</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">±</span><span class="mord">1</span><span class="mclose">)</span></span></span></span></span></p><p>在这里看起来很简单，但计算量还是不小的。</p><p>考虑到 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi><msub><mi>B</mi><mn>1</mn></msub></mrow><annotation encoding="application/x-tex">AB_1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em;"></span><span class="mord mathnormal">A</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.05017em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 是椭圆的焦点弦，我们可以利用焦半径公式。这里就不得不提到圆锥曲线的统一定义了。</p><p><strong>圆锥曲线的统一定义</strong>：到定点 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">F</span></span></span></span> 的距离与到定直线 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>l</mi></mrow><annotation encoding="application/x-tex">l</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord mathnormal" style="margin-right:0.01968em;">l</span></span></span></span> 的距离（<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">F</span></span></span></span> 不在 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>l</mi></mrow><annotation encoding="application/x-tex">l</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord mathnormal" style="margin-right:0.01968em;">l</span></span></span></span> 上）的比 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>e</mi></mrow><annotation encoding="application/x-tex">e</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathnormal">e</span></span></span></span> 是常数的点的轨迹叫作圆锥曲线。</p><p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mtext>如图，椭圆</mtext><mfrac><msup><mi>x</mi><mn>2</mn></msup><msup><mi>a</mi><mn>2</mn></msup></mfrac><mo>+</mo><mfrac><msup><mi>y</mi><mn>2</mn></msup><msup><mi>b</mi><mn>2</mn></msup></mfrac><mo>=</mo><mn>1</mn><mo separator="true">,</mo><msub><mi>F</mi><mn>1</mn></msub><mo separator="true">,</mo><msub><mi>F</mi><mn>2</mn></msub><mtext>为其左右焦点，</mtext><mspace linebreak="newline"></mspace><mtext>直线</mtext><mi>l</mi><mo>:</mo><mi>x</mi><mo>=</mo><mo>−</mo><mfrac><msup><mi>a</mi><mn>2</mn></msup><mi>c</mi></mfrac><mtext>为其左准线,</mtext><mi>A</mi><mi>C</mi><mi mathvariant="normal">⊥</mi><mi>l</mi><mo separator="true">,</mo><mtext>则有</mtext><mo>∣</mo><mi>A</mi><msub><mi>F</mi><mn>1</mn></msub><mo>∣</mo><mo>=</mo><mi>e</mi><mo>∣</mo><mi>A</mi><mi>C</mi><mo>∣</mo></mrow><annotation encoding="application/x-tex">\text{如图，椭圆}\frac{x^2}{a^2}+\frac{y^2}{b^2}=1,F_1,F_2\text{为其左右焦点，}\\\text{直线}l:x=-\frac{a^2}{c}\text{为其左准线,}AC\bot l,\text{则有}\mid AF_1\mid=e\mid AC\mid</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.177108em;vertical-align:-0.686em;"></span><span class="mord text"><span class="mord cjk_fallback">如图，椭圆</span></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.491108em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.740108em;"><span style="top:-2.9890000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:2.177108em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.491108em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">b</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.740108em;"><span style="top:-2.9890000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.8777699999999999em;vertical-align:-0.19444em;"></span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord text"><span class="mord cjk_fallback">为其左右焦点，</span></span></span><span class="mspace newline"></span><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord text"><span class="mord cjk_fallback">直线</span></span><span class="mord mathnormal" style="margin-right:0.01968em;">l</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">:</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:2.177108em;vertical-align:-0.686em;"></span><span class="mord">−</span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.491108em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">c</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord text"><span class="mord cjk_fallback">为其左准线</span><span class="mord">,</span></span><span class="mord mathnormal">A</span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="mord">⊥</span><span class="mord mathnormal" style="margin-right:0.01968em;">l</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord text"><span class="mord cjk_fallback">则有</span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">∣</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">A</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">∣</span></span><span class="base"><span class="strut" style="height:0.36687em;vertical-align:0em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">e</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">∣</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">A</span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">∣</span></span></span></span></span></p><p>实际上这也很好“证明”<s>（证明一个“定义”有种儿子生爸爸矛盾了）</s></p><p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mtext>设点</mtext><mi>A</mi><mo stretchy="false">(</mo><msub><mi>x</mi><mn>0</mn></msub><mo separator="true">,</mo><msub><mi>y</mi><mn>0</mn></msub><mo stretchy="false">)</mo><mo separator="true">,</mo><mtext>则</mtext><mi mathvariant="normal">∣</mi><mi>A</mi><msub><mi>F</mi><mn>1</mn></msub><mi mathvariant="normal">∣</mi><mo>=</mo><msqrt><mrow><mo stretchy="false">(</mo><msub><mi>x</mi><mn>0</mn></msub><mo>+</mo><mi>c</mi><msup><mo stretchy="false">)</mo><mn>2</mn></msup><mo>+</mo><msubsup><mi>y</mi><mn>0</mn><mn>2</mn></msubsup></mrow></msqrt><mo>=</mo><msqrt><mrow><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><mfrac><msup><mi>b</mi><mn>2</mn></msup><msup><mi>a</mi><mn>2</mn></msup></mfrac><mo stretchy="false">)</mo><msubsup><mi>x</mi><mn>0</mn><mn>2</mn></msubsup><mo>+</mo><mn>2</mn><mi>c</mi><msub><mi>x</mi><mn>0</mn></msub><mo>+</mo><msup><mi>c</mi><mn>2</mn></msup><mo>+</mo><msup><mi>b</mi><mn>2</mn></msup></mrow></msqrt><mspace linebreak="newline"></mspace><mo>=</mo><mfrac><mn>1</mn><mi>a</mi></mfrac><msqrt><mrow><msup><mi>c</mi><mn>2</mn></msup><msubsup><mi>x</mi><mn>0</mn><mn>2</mn></msubsup><mo>+</mo><mn>2</mn><mi>c</mi><msup><mi>a</mi><mn>2</mn></msup><msub><mi>x</mi><mn>0</mn></msub><mo>+</mo><msup><mi>a</mi><mn>4</mn></msup></mrow></msqrt><mo>=</mo><mfrac><mn>1</mn><mi>a</mi></mfrac><msqrt><mrow><mo stretchy="false">(</mo><mi>c</mi><msub><mi>x</mi><mn>0</mn></msub><mo>+</mo><msup><mi>a</mi><mn>2</mn></msup><mo stretchy="false">)</mo></mrow></msqrt><mo>=</mo><mi>a</mi><mo>+</mo><mi>e</mi><msub><mi>x</mi><mn>0</mn></msub><mspace linebreak="newline"></mspace><mtext>而</mtext><mi mathvariant="normal">∣</mi><mi>A</mi><mi>C</mi><mi mathvariant="normal">∣</mi><mo>=</mo><msub><mi>x</mi><mn>0</mn></msub><mo>+</mo><mfrac><msup><mi>a</mi><mn>2</mn></msup><mi>c</mi></mfrac><mo>=</mo><mfrac><mrow><mi>e</mi><msub><mi>x</mi><mn>0</mn></msub><mo>+</mo><mi>a</mi></mrow><mi>e</mi></mfrac><mo>=</mo><mfrac><mrow><mi mathvariant="normal">∣</mi><mi>A</mi><msub><mi>F</mi><mn>1</mn></msub><mi mathvariant="normal">∣</mi></mrow><mi>e</mi></mfrac></mrow><annotation encoding="application/x-tex">\text{设点}A(x_0,y_0),\text{则}|AF_1|=\sqrt{(x_0+c)^2+y_0^2}=\sqrt{(1-\frac{b^2}{a^2})x_0^2+2cx_0+c^2+b^2}\\=\frac{1}{a}\sqrt{c^2x_0^2+2ca^2x_0+a^4}=\frac{1}{a}\sqrt{(cx_0+a^2)}=a+ex_0\\\text{而}|AC|=x_0+\frac{a^2}{c}=\frac{ex_0+a}{e}=\frac{|AF_1|}{e}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord text"><span class="mord cjk_fallback">设点</span></span><span class="mord mathnormal">A</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord text"><span class="mord cjk_fallback">则</span></span><span class="mord">∣</span><span class="mord mathnormal">A</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord">∣</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1.84em;vertical-align:-0.5413249999999998em;"></span><span class="mord sqrt"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.2986750000000002em;"><span class="svg-align" style="top:-3.8em;"><span class="pstrut" style="height:3.8em;"></span><span class="mord" style="padding-left:1em;"><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord mathnormal">c</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.740108em;"><span style="top:-2.9890000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7959080000000001em;"><span style="top:-2.433692em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span><span style="top:-3.0448000000000004em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.26630799999999993em;"><span></span></span></span></span></span></span></span></span><span style="top:-3.258675em;"><span class="pstrut" style="height:3.8em;"></span><span class="hide-tail" style="min-width:1.02em;height:1.8800000000000001em;"><svg width='400em' height='1.8800000000000001em' viewBox='0 0 400000 1944' preserveAspectRatio='xMinYMin slice'><path d='M983 90l0 -0c4,-6.7,10,-10,18,-10 H400000v40H1013.1s-83.4,268,-264.1,840c-180.7,572,-277,876.3,-289,913c-4.7,4.7,-12.7,7,-24,7s-12,0,-12,0c-1.3,-3.3,-3.7,-11.7,-7,-25c-35.3,-125.3,-106.7,-373.3,-214,-744c-10,12,-21,25,-33,39s-32,39,-32,39c-6,-5.3,-15,-14,-27,-26s25,-30,25,-30c26.7,-32.7,52,-63,76,-91s52,-60,52,-60s208,722,208,722c56,-175.3,126.3,-397.3,211,-666c84.7,-268.7,153.8,-488.2,207.5,-658.5c53.7,-170.3,84.5,-266.8,92.5,-289.5zM1001 80h400000v40h-400000z'/></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.5413249999999998em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:2.44em;vertical-align:-0.7405709999999996em;"></span><span class="mord sqrt"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.6994290000000003em;"><span class="svg-align" style="top:-4.4em;"><span class="pstrut" style="height:4.4em;"></span><span class="mord" style="padding-left:1em;"><span class="mopen">(</span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.417108em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.740108em;"><span style="top:-2.9890000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">b</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.740108em;"><span style="top:-2.9890000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mclose">)</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7959080000000001em;"><span style="top:-2.433692em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span><span style="top:-3.0448000000000004em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.26630799999999993em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord">2</span><span class="mord mathnormal">c</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord"><span class="mord mathnormal">c</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.740108em;"><span style="top:-2.9890000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord"><span class="mord mathnormal">b</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.740108em;"><span style="top:-2.9890000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span><span style="top:-3.6594290000000007em;"><span class="pstrut" style="height:4.4em;"></span><span class="hide-tail" style="min-width:1.02em;height:2.48em;"><svg width='400em' height='2.48em' viewBox='0 0 400000 2592' preserveAspectRatio='xMinYMin slice'><path d='M424,2478c-1.3,-0.7,-38.5,-172,-111.5,-514c-73,-342,-109.8,-513.3,-110.5,-514c0,-2,-10.7,14.3,-32,49c-4.7,7.3,-9.8,15.7,-15.5,25c-5.7,9.3,-9.8,16,-12.5,20s-5,7,-5,7c-4,-3.3,-8.3,-7.7,-13,-13s-13,-13,-13,-13s76,-122,76,-122s77,-121,77,-121s209,968,209,968c0,-2,84.7,-361.7,254,-1079c169.3,-717.3,254.7,-1077.7,256,-1081l0 -0c4,-6.7,10,-10,18,-10 H400000v40H1014.6s-87.3,378.7,-272.6,1166c-185.3,787.3,-279.3,1182.3,-282,1185c-2,6,-10,9,-24,9c-8,0,-12,-0.7,-12,-2z M1001 80h400000v40h-400000z'/></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.7405709999999996em;"><span></span></span></span></span></span></span><span class="mspace newline"></span><span class="base"><span class="strut" style="height:0.36687em;vertical-align:0em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:2.00744em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.32144em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">a</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord sqrt"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.2986750000000002em;"><span class="svg-align" style="top:-3.8em;"><span class="pstrut" style="height:3.8em;"></span><span class="mord" style="padding-left:1em;"><span class="mord"><span class="mord mathnormal">c</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.740108em;"><span style="top:-2.9890000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7959080000000001em;"><span style="top:-2.433692em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span><span style="top:-3.0448000000000004em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.26630799999999993em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord">2</span><span class="mord mathnormal">c</span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.740108em;"><span style="top:-2.9890000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.740108em;"><span style="top:-2.9890000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">4</span></span></span></span></span></span></span></span></span></span><span style="top:-3.258675em;"><span class="pstrut" style="height:3.8em;"></span><span class="hide-tail" style="min-width:1.02em;height:1.8800000000000001em;"><svg width='400em' height='1.8800000000000001em' viewBox='0 0 400000 1944' preserveAspectRatio='xMinYMin slice'><path d='M983 90l0 -0c4,-6.7,10,-10,18,-10 H400000v40H1013.1s-83.4,268,-264.1,840c-180.7,572,-277,876.3,-289,913c-4.7,4.7,-12.7,7,-24,7s-12,0,-12,0c-1.3,-3.3,-3.7,-11.7,-7,-25c-35.3,-125.3,-106.7,-373.3,-214,-744c-10,12,-21,25,-33,39s-32,39,-32,39c-6,-5.3,-15,-14,-27,-26s25,-30,25,-30c26.7,-32.7,52,-63,76,-91s52,-60,52,-60s208,722,208,722c56,-175.3,126.3,-397.3,211,-666c84.7,-268.7,153.8,-488.2,207.5,-658.5c53.7,-170.3,84.5,-266.8,92.5,-289.5zM1001 80h400000v40h-400000z'/></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.5413249999999998em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:2.00744em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.32144em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">a</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord sqrt"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.983875em;"><span class="svg-align" style="top:-3.2em;"><span class="pstrut" style="height:3.2em;"></span><span class="mord" style="padding-left:1em;"><span class="mopen">(</span><span class="mord mathnormal">c</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.740108em;"><span style="top:-2.9890000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mclose">)</span></span></span><span style="top:-2.9438750000000002em;"><span class="pstrut" style="height:3.2em;"></span><span class="hide-tail" style="min-width:1.02em;height:1.28em;"><svg width='400em' height='1.28em' viewBox='0 0 400000 1296' preserveAspectRatio='xMinYMin slice'><path d='M263,681c0.7,0,18,39.7,52,119c34,79.3,68.167,158.7,102.5,238c34.3,79.3,51.8,119.3,52.5,120c340,-704.7,510.7,-1060.3,512,-1067l0 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style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.58056em;vertical-align:-0.15em;"></span><span class="mord mathnormal">e</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span><span class="mspace newline"></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord text"><span class="mord cjk_fallback">而</span></span><span class="mord">∣</span><span class="mord mathnormal">A</span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="mord">∣</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.73333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:2.177108em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.491108em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">c</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1.9463300000000001em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.2603300000000002em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">e</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">e</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord mathnormal">a</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:2.113em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.427em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">e</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">∣</span><span class="mord mathnormal">A</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord">∣</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></p><p><img src="https://cdn.jsdelivr.net/gh/Yurchiu/PicGo/3d233a1900a8ee269130546960088a14.png" alt="" /></p><p>一般来说课本内容或一些资料就到此为止了，而我们学习抛物线时会有这么一个结论：</p><p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>A</mi><mi>B</mi><mtext>是过抛物线</mtext><msup><mi>y</mi><mn>2</mn></msup><mo>=</mo><mn>2</mn><mi>p</mi><mi>x</mi><mtext>的一条直线，F为抛物线焦点，</mtext><mspace linebreak="newline"></mspace><mtext>A在x轴上方，</mtext><mi>θ</mi><mtext>为直线AB的倾斜角，则有</mtext><mspace linebreak="newline"></mspace><mi mathvariant="normal">∣</mi><mi>A</mi><mi>F</mi><mi mathvariant="normal">∣</mi><mo>=</mo><mfrac><mi>p</mi><mrow><mn>1</mn><mo>−</mo><mi>cos</mi><mo>⁡</mo><mi>θ</mi></mrow></mfrac><mo separator="true">,</mo><mi mathvariant="normal">∣</mi><mi>B</mi><mi>F</mi><mi mathvariant="normal">∣</mi><mo>=</mo><mfrac><mi>p</mi><mrow><mn>1</mn><mo>+</mo><mi>cos</mi><mo>⁡</mo><mi>θ</mi></mrow></mfrac><mspace linebreak="newline"></mspace><mi mathvariant="normal">∣</mi><mi>A</mi><mi>B</mi><mi mathvariant="normal">∣</mi><mo>=</mo><mfrac><mrow><mn>2</mn><mi>p</mi></mrow><mrow><msup><mo><mi>sin</mi><mo>⁡</mo></mo><mn>2</mn></msup><mi>θ</mi></mrow></mfrac><mo separator="true">,</mo><mfrac><mn>1</mn><mrow><mi mathvariant="normal">∣</mi><mi>A</mi><mi>F</mi><mi mathvariant="normal">∣</mi></mrow></mfrac><mo>+</mo><mfrac><mn>1</mn><mrow><mi mathvariant="normal">∣</mi><mi>B</mi><mi>F</mi><mi mathvariant="normal">∣</mi></mrow></mfrac><mo>=</mo><mfrac><mn>2</mn><mi>p</mi></mfrac></mrow><annotation encoding="application/x-tex">AB\text{是过抛物线}y^2=2px\text{的一条直线，F为抛物线焦点，}\\\text{A在x轴上方，}\theta\text{为直线AB的倾斜角，则有}\\|AF|=\frac{p}{1-\cos\theta},|BF|=\frac{p}{1+\cos\theta}\\|AB|=\frac{2p}{\sin^2\theta},\frac{1}{|AF|}+\frac{1}{|BF|}=\frac{2}{p}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0585479999999998em;vertical-align:-0.19444em;"></span><span class="mord mathnormal">A</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mord text"><span class="mord cjk_fallback">是过抛物线</span></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641079999999999em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.8777699999999999em;vertical-align:-0.19444em;"></span><span class="mord">2</span><span class="mord mathnormal">p</span><span class="mord mathnormal">x</span><span class="mord text"><span class="mord cjk_fallback">的一条直线，</span><span class="mord">F</span><span class="mord cjk_fallback">为抛物线焦点，</span></span></span><span class="mspace newline"></span><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord text"><span class="mord">A</span><span class="mord cjk_fallback">在</span><span class="mord">x</span><span class="mord cjk_fallback">轴上方，</span></span><span class="mord mathnormal" style="margin-right:0.02778em;">θ</span><span class="mord text"><span class="mord cjk_fallback">为直线</span><span class="mord">AB</span><span class="mord cjk_fallback">的倾斜角，则有</span></span></span><span class="mspace newline"></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">∣</span><span class="mord mathnormal">A</span><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="mord">∣</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1.87689em;vertical-align:-0.7693300000000001em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.1075599999999999em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mop">cos</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">θ</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">p</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.7693300000000001em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">∣</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="mord">∣</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1.87689em;vertical-align:-0.7693300000000001em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.1075599999999999em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mop">cos</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">θ</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">p</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.7693300000000001em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span><span class="mspace newline"></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">∣</span><span class="mord mathnormal">A</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mord">∣</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:2.25744em;vertical-align:-0.936em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.32144em;"><span style="top:-2.2381320000000002em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mop"><span class="mop">sin</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.871868em;"><span style="top:-3.12076em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">θ</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span><span class="mord mathnormal">p</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.761868em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.32144em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">∣</span><span class="mord mathnormal">A</span><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="mord">∣</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.936em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:2.25744em;vertical-align:-0.936em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.32144em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">∣</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="mord">∣</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.936em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:2.20188em;vertical-align:-0.8804400000000001em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.32144em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">p</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.8804400000000001em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></p><p>那么是不是所有圆锥曲线都有类似性质呢，根据圆锥曲线统一定义我们推断这肯定是存在的。</p><p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mtext>如上图，对于一个椭圆，设AB的倾斜角为</mtext><mi>α</mi><mo separator="true">,</mo><mi mathvariant="normal">∣</mi><mi>A</mi><msub><mi>F</mi><mn>1</mn></msub><mi mathvariant="normal">∣</mi><mo>=</mo><mi>m</mi><mo separator="true">,</mo><mtext>则</mtext><mi mathvariant="normal">∣</mi><mi>A</mi><mi>E</mi><mi mathvariant="normal">∣</mi><mo>=</mo><mi>m</mi><mi>cos</mi><mo>⁡</mo><mi>α</mi><mspace linebreak="newline"></mspace><mi mathvariant="normal">∣</mi><mi>A</mi><mi>C</mi><mi mathvariant="normal">∣</mi><mo>=</mo><mfrac><msup><mi>a</mi><mn>2</mn></msup><mi>c</mi></mfrac><mo>−</mo><mi>c</mi><mo>+</mo><mi>m</mi><mi>cos</mi><mo>⁡</mo><mi>α</mi><mo>=</mo><mfrac><mrow><mi>a</mi><mi>m</mi></mrow><mi>c</mi></mfrac><mo>⟹</mo><mi>m</mi><mo>=</mo><mfrac><msup><mi>b</mi><mn>2</mn></msup><mrow><mi>a</mi><mo>−</mo><mi>c</mi><mi>cos</mi><mo>⁡</mo><mi>α</mi></mrow></mfrac><mo separator="true">,</mo><mtext>即</mtext><mi mathvariant="normal">∣</mi><mi>A</mi><msub><mi>F</mi><mn>1</mn></msub><mi mathvariant="normal">∣</mi><mo>=</mo><mfrac><msup><mi>b</mi><mn>2</mn></msup><mrow><mi>a</mi><mo>−</mo><mi>c</mi><mi>cos</mi><mo>⁡</mo><mi>α</mi></mrow></mfrac><mtext>同理可得</mtext><mi mathvariant="normal">∣</mi><mi>B</mi><msub><mi>F</mi><mn>1</mn></msub><mi mathvariant="normal">∣</mi><mo>=</mo><mfrac><msup><mi>b</mi><mn>2</mn></msup><mrow><mi>a</mi><mo>+</mo><mi>c</mi><mi>cos</mi><mo>⁡</mo><mi>α</mi></mrow></mfrac><mspace linebreak="newline"></mspace><mi mathvariant="normal">∣</mi><mi>A</mi><mi>B</mi><mi mathvariant="normal">∣</mi><mo>=</mo><mfrac><mrow><mn>2</mn><mi>a</mi><mi>b</mi></mrow><mrow><msup><mi>a</mi><mn>2</mn></msup><mo>−</mo><msup><mi>c</mi><mn>2</mn></msup><msup><mo><mi>cos</mi><mo>⁡</mo></mo><mn>2</mn></msup><mi>α</mi></mrow></mfrac><mo separator="true">,</mo><mfrac><mn>1</mn><mrow><mi mathvariant="normal">∣</mi><mi>A</mi><msub><mi>F</mi><mn>1</mn></msub><mi mathvariant="normal">∣</mi></mrow></mfrac><mo>+</mo><mfrac><mn>1</mn><mrow><mi mathvariant="normal">∣</mi><mi>B</mi><msub><mi>F</mi><mn>1</mn></msub><mi mathvariant="normal">∣</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><mn>2</mn><mi>a</mi></mrow><msup><mi>b</mi><mn>2</mn></msup></mfrac><mspace linebreak="newline"></mspace><mtext>同理，对于双曲线</mtext><mfrac><msup><mi>x</mi><mn>2</mn></msup><msup><mi>a</mi><mn>2</mn></msup></mfrac><mo>−</mo><mfrac><msup><mi>y</mi><mn>2</mn></msup><msup><mi>b</mi><mn>2</mn></msup></mfrac><mo>=</mo><mn>1</mn><mo separator="true">,</mo><mtext>过其左焦点F的直线AB，有</mtext><mspace linebreak="newline"></mspace><mi mathvariant="normal">∣</mi><mi>A</mi><mi>F</mi><mi mathvariant="normal">∣</mi><mo>=</mo><mfrac><msup><mi>b</mi><mn>2</mn></msup><mrow><mi>a</mi><mo>+</mo><mi>c</mi><mi>cos</mi><mo>⁡</mo><mi>α</mi></mrow></mfrac><mo separator="true">,</mo><mi mathvariant="normal">∣</mi><mi>B</mi><mi>F</mi><mi mathvariant="normal">∣</mi><mo>=</mo><mfrac><msup><mi>b</mi><mn>2</mn></msup><mrow><mi>a</mi><mo>−</mo><mi>c</mi><mi>cos</mi><mo>⁡</mo><mi>α</mi></mrow></mfrac><mo separator="true">,</mo><mi mathvariant="normal">∣</mi><mi>A</mi><mi>B</mi><mi mathvariant="normal">∣</mi><mo>=</mo><mfrac><mrow><mn>2</mn><mi>a</mi><mi>b</mi></mrow><mrow><msup><mi>a</mi><mn>2</mn></msup><mo>−</mo><msup><mi>c</mi><mn>2</mn></msup><msup><mo><mi>cos</mi><mo>⁡</mo></mo><mn>2</mn></msup><mi>α</mi></mrow></mfrac><mo separator="true">,</mo><mfrac><mn>1</mn><mrow><mi mathvariant="normal">∣</mi><mi>A</mi><msub><mi>F</mi><mn>1</mn></msub><mi mathvariant="normal">∣</mi></mrow></mfrac><mo>+</mo><mfrac><mn>1</mn><mrow><mi mathvariant="normal">∣</mi><mi>B</mi><msub><mi>F</mi><mn>1</mn></msub><mi mathvariant="normal">∣</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><mn>2</mn><mi>a</mi></mrow><msup><mi>b</mi><mn>2</mn></msup></mfrac><mspace linebreak="newline"></mspace><mtext>注意AB交于双曲线左支，所以</mtext><mi mathvariant="normal">∣</mi><mi>cos</mi><mo>⁡</mo><mi>α</mi><mi mathvariant="normal">∣</mi><mo>≤</mo><mfrac><mi>a</mi><mi>c</mi></mfrac></mrow><annotation encoding="application/x-tex">\text{如上图，对于一个椭圆，设AB的倾斜角为}\alpha,|AF_1|=m,\text{则}|AE|=m\cos\alpha\\|AC|=\frac{a^2}{c}-c+m\cos\alpha=\frac{am}{c}\Longrightarrow m=\frac{b^2}{a-c\cos\alpha},\text{即}|AF_1|=\frac{b^2}{a-c\cos\alpha}\text{同理可得}|BF_1|=\frac{b^2}{a+c\cos\alpha}\\|AB|=\frac{2ab}{a^2-c^2\cos^2\alpha},\frac{1}{|AF_1|}+\frac{1}{|BF_1|}=\frac{2a}{b^2}\\\text{同理，对于双曲线}\frac{x^2}{a^2}-\frac{y^2}{b^2}=1,\text{过其左焦点F的直线AB，有}\\|AF|=\frac{b^2}{a+c\cos\alpha},|BF|=\frac{b^2}{a-c\cos\alpha},|AB|=\frac{2ab}{a^2-c^2\cos^2\alpha},\frac{1}{|AF_1|}+\frac{1}{|BF_1|}=\frac{2a}{b^2}\\\text{注意AB交于双曲线左支，所以}|\cos\alpha|\leq\frac{a}{c}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord text"><span class="mord cjk_fallback">如上图，对于一个椭圆，设</span><span class="mord">AB</span><span class="mord cjk_fallback">的倾斜角为</span></span><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">∣</span><span class="mord mathnormal">A</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord">∣</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">m</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord text"><span class="mord cjk_fallback">则</span></span><span class="mord">∣</span><span class="mord mathnormal">A</span><span class="mord mathnormal" style="margin-right:0.05764em;">E</span><span class="mord">∣</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathnormal">m</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mop">cos</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal" style="margin-right:0.0037em;">α</span></span><span class="mspace newline"></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">∣</span><span class="mord mathnormal">A</span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="mord">∣</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:2.177108em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.491108em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">c</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.66666em;vertical-align:-0.08333em;"></span><span class="mord mathnormal">c</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathnormal">m</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mop">cos</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1.7935600000000003em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.10756em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">c</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="mord mathnormal">m</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">⟹</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathnormal">m</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:2.260438em;vertical-align:-0.7693300000000001em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.491108em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord mathnormal">c</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mop">cos</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal" style="margin-right:0.0037em;">α</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">b</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.7693300000000001em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord text"><span class="mord cjk_fallback">即</span></span><span class="mord">∣</span><span class="mord mathnormal">A</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord">∣</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:2.260438em;vertical-align:-0.7693300000000001em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.491108em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord mathnormal">c</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mop">cos</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal" style="margin-right:0.0037em;">α</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">b</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.7693300000000001em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord text"><span class="mord cjk_fallback">同理可得</span></span><span class="mord">∣</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord">∣</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:2.260438em;vertical-align:-0.7693300000000001em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.491108em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord mathnormal">c</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mop">cos</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal" style="margin-right:0.0037em;">α</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">b</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.7693300000000001em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span><span class="mspace newline"></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">∣</span><span class="mord mathnormal">A</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mord">∣</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:2.30744em;vertical-align:-0.936em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.37144em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.740108em;"><span style="top:-2.9890000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord"><span class="mord mathnormal">c</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.740108em;"><span style="top:-2.9890000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mop"><span class="mop">cos</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.740108em;"><span style="top:-2.9890000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal" style="margin-right:0.0037em;">α</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span><span class="mord mathnormal">a</span><span class="mord mathnormal">b</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.7693300000000001em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.32144em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">∣</span><span class="mord mathnormal">A</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord">∣</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.936em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:2.25744em;vertical-align:-0.936em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.32144em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">∣</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord">∣</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.936em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:2.00744em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.32144em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">b</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.740108em;"><span style="top:-2.9890000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span><span class="mord mathnormal">a</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span><span class="mspace newline"></span><span class="base"><span class="strut" style="height:2.177108em;vertical-align:-0.686em;"></span><span class="mord text"><span class="mord cjk_fallback">同理，对于双曲线</span></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.491108em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.740108em;"><span style="top:-2.9890000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:2.177108em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.491108em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">b</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.740108em;"><span style="top:-2.9890000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.8777699999999999em;vertical-align:-0.19444em;"></span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord text"><span class="mord cjk_fallback">过其左焦点</span><span class="mord">F</span><span class="mord cjk_fallback">的直线</span><span class="mord">AB</span><span class="mord cjk_fallback">，有</span></span></span><span class="mspace newline"></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">∣</span><span class="mord mathnormal">A</span><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="mord">∣</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:2.260438em;vertical-align:-0.7693300000000001em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.491108em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord mathnormal">c</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mop">cos</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal" style="margin-right:0.0037em;">α</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">b</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.7693300000000001em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">∣</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="mord">∣</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:2.260438em;vertical-align:-0.7693300000000001em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.491108em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord mathnormal">c</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mop">cos</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal" style="margin-right:0.0037em;">α</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">b</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.7693300000000001em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">∣</span><span class="mord mathnormal">A</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mord">∣</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:2.30744em;vertical-align:-0.936em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.37144em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.740108em;"><span style="top:-2.9890000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord"><span class="mord mathnormal">c</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.740108em;"><span style="top:-2.9890000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mop"><span class="mop">cos</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.740108em;"><span style="top:-2.9890000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal" style="margin-right:0.0037em;">α</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span><span class="mord mathnormal">a</span><span class="mord mathnormal">b</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.7693300000000001em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.32144em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">∣</span><span class="mord mathnormal">A</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord">∣</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.936em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:2.25744em;vertical-align:-0.936em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.32144em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">∣</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord">∣</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.936em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:2.00744em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.32144em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">b</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.740108em;"><span style="top:-2.9890000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span><span class="mord mathnormal">a</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span><span class="mspace newline"></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord text"><span class="mord cjk_fallback">注意</span><span class="mord">AB</span><span class="mord cjk_fallback">交于双曲线左支，所以</span></span><span class="mord">∣</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mop">cos</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="mord">∣</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1.7935600000000003em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.10756em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">c</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">a</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></p><p>利用该结论，再看上面那道题：</p><p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mtext>设</mtext><msub><mi>F</mi><mn>1</mn></msub><mo separator="true">,</mo><msub><mi>F</mi><mn>2</mn></msub><mtext>分别为椭圆</mtext><mfrac><msup><mi>x</mi><mn>2</mn></msup><mn>3</mn></mfrac><mo>+</mo><msup><mi>y</mi><mn>2</mn></msup><mo>=</mo><mn>1</mn><mtext>的左、右焦点，</mtext><mspace linebreak="newline"></mspace><mtext>点</mtext><mi>A</mi><mo separator="true">,</mo><mi>B</mi><mtext>在椭圆上，若</mtext><mover accent="true"><mrow><msub><mi>F</mi><mn>1</mn></msub><mi>A</mi></mrow><mo stretchy="true">→</mo></mover><mo>=</mo><mn>5</mn><mover accent="true"><mrow><msub><mi>F</mi><mn>2</mn></msub><mi>B</mi></mrow><mo stretchy="true">→</mo></mover><mo separator="true">,</mo><mtext>则点A的坐标是</mtext><mo stretchy="false">(</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\text{设}F_1,F_2\text{分别为椭圆}\frac{x^2}{3}+y^2=1\text{的左、右焦点，}\\\text{点}A,B\text{在椭圆上，若}\overrightarrow{F_1A}=5\overrightarrow{F_2B},\text{则点A的坐标是}()</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.177108em;vertical-align:-0.686em;"></span><span class="mord text"><span class="mord cjk_fallback">设</span></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord text"><span class="mord cjk_fallback">分别为椭圆</span></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.491108em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">3</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1.0585479999999998em;vertical-align:-0.19444em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641079999999999em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord">1</span><span class="mord text"><span class="mord cjk_fallback">的左、右焦点，</span></span></span><span class="mspace newline"></span><span class="base"><span class="strut" style="height:1.39977em;vertical-align:-0.19444em;"></span><span class="mord text"><span class="mord cjk_fallback">点</span></span><span class="mord mathnormal">A</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mord text"><span class="mord cjk_fallback">在椭圆上，若</span></span><span class="mord accent"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.20533em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord mathnormal">A</span></span></span><span class="svg-align" style="top:-3.6833299999999998em;"><span class="pstrut" style="height:3em;"></span><span class="hide-tail" style="height:0.522em;min-width:0.888em;"><svg width='400em' height='0.522em' viewBox='0 0 400000 522' preserveAspectRatio='xMaxYMin slice'><path d='M0 241v40h399891c-47.3 35.3-84 78-110 128-16.7 32-27.7 63.7-33 95 0 1.3-.2 2.7-.5 4-.3 1.3-.5 2.3-.5 3 0 7.3 6.7 11 20 11 8 0 13.2-.8 15.5-2.5 2.3-1.7 4.2-5.5 5.5-11.5 2-13.3 5.7-27 11-41 14.7-44.7 39-84.5 73-119.5s73.7-60.2 119-75.5c6-2 9-5.7 9-11s-3-9-9-11c-45.3-15.3-85-40.5-119-75.5s-58.3-74.8-73-119.5c-4.7-14-8.3-27.3-11-40-1.3-6.7-3.2-10.8-5.5-12.5-2.3-1.7-7.5-2.5-15.5-2.5-14 0-21 3.7-21 11 0 2 2 10.3 6 25 20.7 83.3 67 151.7 139 205zm0 0v40h399900v-40z'/></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1.45533em;vertical-align:-0.25em;"></span><span class="mord">5</span><span class="mord accent"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.20533em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span><span class="svg-align" style="top:-3.6833299999999998em;"><span class="pstrut" style="height:3em;"></span><span class="hide-tail" style="height:0.522em;min-width:0.888em;"><svg width='400em' height='0.522em' viewBox='0 0 400000 522' preserveAspectRatio='xMaxYMin slice'><path d='M0 241v40h399891c-47.3 35.3-84 78-110 128-16.7 32-27.7 63.7-33 95 0 1.3-.2 2.7-.5 4-.3 1.3-.5 2.3-.5 3 0 7.3 6.7 11 20 11 8 0 13.2-.8 15.5-2.5 2.3-1.7 4.2-5.5 5.5-11.5 2-13.3 5.7-27 11-41 14.7-44.7 39-84.5 73-119.5s73.7-60.2 119-75.5c6-2 9-5.7 9-11s-3-9-9-11c-45.3-15.3-85-40.5-119-75.5s-58.3-74.8-73-119.5c-4.7-14-8.3-27.3-11-40-1.3-6.7-3.2-10.8-5.5-12.5-2.3-1.7-7.5-2.5-15.5-2.5-14 0-21 3.7-21 11 0 2 2 10.3 6 25 20.7 83.3 67 151.7 139 205zm0 0v40h399900v-40z'/></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord text"><span class="mord cjk_fallback">则点</span><span class="mord">A</span><span class="mord cjk_fallback">的坐标是</span></span><span class="mopen">(</span><span class="mclose">)</span></span></span></span></span></p><p><strong>解法二：</strong></p><p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi mathvariant="normal">∣</mi><msub><mi>F</mi><mn>1</mn></msub><mi>A</mi><mi mathvariant="normal">∣</mi><mo>=</mo><mn>5</mn><mi mathvariant="normal">∣</mi><msub><mi>F</mi><mn>1</mn></msub><msub><mi>B</mi><mn>1</mn></msub><mi mathvariant="normal">∣</mi><mo>⟹</mo><mfrac><mn>1</mn><mrow><msqrt><mn>3</mn></msqrt><mo>−</mo><msqrt><mn>2</mn></msqrt><mi>cos</mi><mo>⁡</mo><mi>θ</mi></mrow></mfrac><mo>=</mo><mfrac><mn>5</mn><mrow><msqrt><mn>3</mn></msqrt><mo>+</mo><msqrt><mn>2</mn></msqrt><mi>cos</mi><mo>⁡</mo><mi>θ</mi></mrow></mfrac><mo>⟹</mo><mi>cos</mi><mo>⁡</mo><mi>θ</mi><mo>=</mo><mfrac><msqrt><mn>6</mn></msqrt><mn>3</mn></mfrac><mo>=</mo><mfrac><mi>c</mi><mi>a</mi></mfrac><mspace linebreak="newline"></mspace><mo>∴</mo><mi>A</mi><mo stretchy="false">(</mo><mn>0</mn><mo separator="true">,</mo><mo>±</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">|F_1A|=5|F_1B_1|\Longrightarrow\frac{1}{\sqrt{3}-\sqrt{2}\cos\theta}=\frac{5}{\sqrt{3}+\sqrt{2}\cos\theta}\Longrightarrow\cos\theta=\frac{\sqrt{6}}{3}=\frac{c}{a}\\\therefore A(0,\pm1)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">∣</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord mathnormal">A</span><span class="mord">∣</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">5</span><span class="mord">∣</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.05017em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord">∣</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">⟹</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:2.25144em;vertical-align:-0.93em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.32144em;"><span style="top:-2.2027799999999997em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord sqrt"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.90722em;"><span class="svg-align" style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord" style="padding-left:0.833em;"><span class="mord">3</span></span></span><span style="top:-2.86722em;"><span class="pstrut" style="height:3em;"></span><span class="hide-tail" style="min-width:0.853em;height:1.08em;"><svg width='400em' height='1.08em' viewBox='0 0 400000 1080' preserveAspectRatio='xMinYMin slice'><path d='M95,702c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z'/></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.13278em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord sqrt"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.90722em;"><span class="svg-align" style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord" style="padding-left:0.833em;"><span class="mord">2</span></span></span><span style="top:-2.86722em;"><span class="pstrut" style="height:3em;"></span><span class="hide-tail" style="min-width:0.853em;height:1.08em;"><svg width='400em' height='1.08em' viewBox='0 0 400000 1080' preserveAspectRatio='xMinYMin slice'><path d='M95,702c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z'/></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.13278em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mop">cos</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">θ</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.93em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:2.25144em;vertical-align:-0.93em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.32144em;"><span style="top:-2.2027799999999997em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord sqrt"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.90722em;"><span class="svg-align" style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord" style="padding-left:0.833em;"><span class="mord">3</span></span></span><span style="top:-2.86722em;"><span class="pstrut" style="height:3em;"></span><span class="hide-tail" style="min-width:0.853em;height:1.08em;"><svg width='400em' height='1.08em' viewBox='0 0 400000 1080' preserveAspectRatio='xMinYMin slice'><path d='M95,702c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z'/></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.13278em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord sqrt"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.90722em;"><span class="svg-align" style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord" style="padding-left:0.833em;"><span class="mord">2</span></span></span><span style="top:-2.86722em;"><span class="pstrut" style="height:3em;"></span><span class="hide-tail" style="min-width:0.853em;height:1.08em;"><svg width='400em' height='1.08em' viewBox='0 0 400000 1080' preserveAspectRatio='xMinYMin slice'><path d='M95,702c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z'/></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.13278em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mop">cos</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">θ</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">5</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.93em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">⟹</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mop">cos</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">θ</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:2.27022em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.5842200000000002em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">3</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord sqrt"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.90722em;"><span class="svg-align" style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord" style="padding-left:0.833em;"><span class="mord">6</span></span></span><span style="top:-2.86722em;"><span class="pstrut" style="height:3em;"></span><span class="hide-tail" style="min-width:0.853em;height:1.08em;"><svg width='400em' height='1.08em' viewBox='0 0 400000 1080' preserveAspectRatio='xMinYMin slice'><path d='M95,702c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z'/></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.13278em;"><span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1.7935600000000003em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.10756em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">a</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">c</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span><span class="mspace newline"></span><span class="base"><span class="strut" style="height:0.69224em;vertical-align:0em;"></span><span class="mrel amsrm">∴</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">A</span><span class="mopen">(</span><span class="mord">0</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">±</span><span class="mord">1</span><span class="mclose">)</span></span></span></span></span></p><p>这样计算甚至降到了口算量级，下面我们再来看这样一道题。</p><p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mtext>设点P为椭圆C：</mtext><mfrac><msup><mi>x</mi><mn>2</mn></msup><mn>6</mn></mfrac><mo>+</mo><mfrac><msup><mi>y</mi><mn>2</mn></msup><mn>2</mn></mfrac><mo>=</mo><mn>1</mn><mtext>上任意一点，过两焦点</mtext><msub><mi>F</mi><mn>1</mn></msub><mo separator="true">,</mo><msub><mi>F</mi><mn>2</mn></msub><mtext>的弦分别为PA,PB,设</mtext><mspace linebreak="newline"></mspace><mover accent="true"><mrow><mi>P</mi><msub><mi>F</mi><mn>1</mn></msub></mrow><mo stretchy="true">→</mo></mover><mo>=</mo><msub><mi>λ</mi><mn>1</mn></msub><mover accent="true"><mrow><msub><mi>F</mi><mn>1</mn></msub><mi>A</mi></mrow><mo stretchy="true">→</mo></mover><mo separator="true">,</mo><mover accent="true"><mrow><mi>P</mi><msub><mi>F</mi><mn>2</mn></msub></mrow><mo stretchy="true">→</mo></mover><mo>=</mo><msub><mi>λ</mi><mn>2</mn></msub><mover accent="true"><mrow><msub><mi>F</mi><mn>2</mn></msub><mi>B</mi></mrow><mo stretchy="true">→</mo></mover><mo separator="true">,</mo><mtext>问</mtext><msub><mi>λ</mi><mn>1</mn></msub><mo>+</mo><msub><mi>λ</mi><mn>2</mn></msub><mtext>是否是定值，并证明</mtext></mrow><annotation encoding="application/x-tex">\text{设点P为椭圆C：}\frac{x^2}{6}+\frac{y^2}2=1\text{上任意一点，过两焦点}F_1,F_2\text{的弦分别为PA,PB,设}\\\overrightarrow{PF_1}=\lambda_1\overrightarrow{F_1A},\overrightarrow{PF_2}=\lambda_2\overrightarrow{F_2B},\text{问}\lambda_1+\lambda_2\text{是否是定值，并证明}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.177108em;vertical-align:-0.686em;"></span><span class="mord text"><span class="mord cjk_fallback">设点</span><span class="mord">P</span><span class="mord cjk_fallback">为椭圆</span><span class="mord">C</span><span class="mord cjk_fallback">：</span></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.491108em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">6</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:2.177108em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.491108em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord">2</span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.8777699999999999em;vertical-align:-0.19444em;"></span><span class="mord">1</span><span class="mord text"><span class="mord cjk_fallback">上任意一点，过两焦点</span></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord text"><span class="mord cjk_fallback">的弦分别为</span><span class="mord">PA,PB,</span><span class="mord cjk_fallback">设</span></span></span><span class="mspace newline"></span><span class="base"><span class="strut" style="height:1.35533em;vertical-align:-0.15em;"></span><span class="mord accent"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.20533em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span><span class="svg-align" style="top:-3.6833299999999998em;"><span class="pstrut" style="height:3em;"></span><span class="hide-tail" style="height:0.522em;min-width:0.888em;"><svg width='400em' height='0.522em' viewBox='0 0 400000 522' preserveAspectRatio='xMaxYMin slice'><path d='M0 241v40h399891c-47.3 35.3-84 78-110 128-16.7 32-27.7 63.7-33 95 0 1.3-.2 2.7-.5 4-.3 1.3-.5 2.3-.5 3 0 7.3 6.7 11 20 11 8 0 13.2-.8 15.5-2.5 2.3-1.7 4.2-5.5 5.5-11.5 2-13.3 5.7-27 11-41 14.7-44.7 39-84.5 73-119.5s73.7-60.2 119-75.5c6-2 9-5.7 9-11s-3-9-9-11c-45.3-15.3-85-40.5-119-75.5s-58.3-74.8-73-119.5c-4.7-14-8.3-27.3-11-40-1.3-6.7-3.2-10.8-5.5-12.5-2.3-1.7-7.5-2.5-15.5-2.5-14 0-21 3.7-21 11 0 2 2 10.3 6 25 20.7 83.3 67 151.7 139 205zm0 0v40h399900v-40z'/></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1.39977em;vertical-align:-0.19444em;"></span><span class="mord"><span class="mord mathnormal">λ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord accent"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.20533em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord mathnormal">A</span></span></span><span class="svg-align" style="top:-3.6833299999999998em;"><span class="pstrut" style="height:3em;"></span><span class="hide-tail" style="height:0.522em;min-width:0.888em;"><svg width='400em' height='0.522em' viewBox='0 0 400000 522' preserveAspectRatio='xMaxYMin slice'><path d='M0 241v40h399891c-47.3 35.3-84 78-110 128-16.7 32-27.7 63.7-33 95 0 1.3-.2 2.7-.5 4-.3 1.3-.5 2.3-.5 3 0 7.3 6.7 11 20 11 8 0 13.2-.8 15.5-2.5 2.3-1.7 4.2-5.5 5.5-11.5 2-13.3 5.7-27 11-41 14.7-44.7 39-84.5 73-119.5s73.7-60.2 119-75.5c6-2 9-5.7 9-11s-3-9-9-11c-45.3-15.3-85-40.5-119-75.5s-58.3-74.8-73-119.5c-4.7-14-8.3-27.3-11-40-1.3-6.7-3.2-10.8-5.5-12.5-2.3-1.7-7.5-2.5-15.5-2.5-14 0-21 3.7-21 11 0 2 2 10.3 6 25 20.7 83.3 67 151.7 139 205zm0 0v40h399900v-40z'/></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord accent"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.20533em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span><span class="svg-align" style="top:-3.6833299999999998em;"><span class="pstrut" style="height:3em;"></span><span class="hide-tail" style="height:0.522em;min-width:0.888em;"><svg width='400em' height='0.522em' viewBox='0 0 400000 522' preserveAspectRatio='xMaxYMin slice'><path d='M0 241v40h399891c-47.3 35.3-84 78-110 128-16.7 32-27.7 63.7-33 95 0 1.3-.2 2.7-.5 4-.3 1.3-.5 2.3-.5 3 0 7.3 6.7 11 20 11 8 0 13.2-.8 15.5-2.5 2.3-1.7 4.2-5.5 5.5-11.5 2-13.3 5.7-27 11-41 14.7-44.7 39-84.5 73-119.5s73.7-60.2 119-75.5c6-2 9-5.7 9-11s-3-9-9-11c-45.3-15.3-85-40.5-119-75.5s-58.3-74.8-73-119.5c-4.7-14-8.3-27.3-11-40-1.3-6.7-3.2-10.8-5.5-12.5-2.3-1.7-7.5-2.5-15.5-2.5-14 0-21 3.7-21 11 0 2 2 10.3 6 25 20.7 83.3 67 151.7 139 205zm0 0v40h399900v-40z'/></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1.39977em;vertical-align:-0.19444em;"></span><span class="mord"><span class="mord mathnormal">λ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord accent"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.20533em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span><span class="svg-align" style="top:-3.6833299999999998em;"><span class="pstrut" style="height:3em;"></span><span class="hide-tail" style="height:0.522em;min-width:0.888em;"><svg width='400em' height='0.522em' viewBox='0 0 400000 522' preserveAspectRatio='xMaxYMin slice'><path d='M0 241v40h399891c-47.3 35.3-84 78-110 128-16.7 32-27.7 63.7-33 95 0 1.3-.2 2.7-.5 4-.3 1.3-.5 2.3-.5 3 0 7.3 6.7 11 20 11 8 0 13.2-.8 15.5-2.5 2.3-1.7 4.2-5.5 5.5-11.5 2-13.3 5.7-27 11-41 14.7-44.7 39-84.5 73-119.5s73.7-60.2 119-75.5c6-2 9-5.7 9-11s-3-9-9-11c-45.3-15.3-85-40.5-119-75.5s-58.3-74.8-73-119.5c-4.7-14-8.3-27.3-11-40-1.3-6.7-3.2-10.8-5.5-12.5-2.3-1.7-7.5-2.5-15.5-2.5-14 0-21 3.7-21 11 0 2 2 10.3 6 25 20.7 83.3 67 151.7 139 205zm0 0v40h399900v-40z'/></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord text"><span class="mord cjk_fallback">问</span></span><span class="mord"><span class="mord mathnormal">λ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.84444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">λ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord text"><span class="mord cjk_fallback">是否是定值，并证明</span></span></span></span></span></span></p><p>这里我们直接利用焦点弦结论</p><p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mo>∵</mo><mfrac><mn>1</mn><mrow><mi mathvariant="normal">∣</mi><mi>A</mi><msub><mi>F</mi><mn>1</mn></msub><mi mathvariant="normal">∣</mi></mrow></mfrac><mo>+</mo><mfrac><mn>1</mn><mrow><mi mathvariant="normal">∣</mi><mi>P</mi><msub><mi>F</mi><mn>1</mn></msub><mi mathvariant="normal">∣</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><mn>1</mn><mo>+</mo><msub><mi>λ</mi><mn>1</mn></msub></mrow><mrow><mi mathvariant="normal">∣</mi><mi>P</mi><msub><mi>F</mi><mn>1</mn></msub><mi mathvariant="normal">∣</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><mn>2</mn><mi>a</mi></mrow><msup><mi>b</mi><mn>2</mn></msup></mfrac><mo>=</mo><msqrt><mn>6</mn></msqrt><mo separator="true">,</mo><mfrac><mn>1</mn><mrow><mi mathvariant="normal">∣</mi><mi>A</mi><msub><mi>F</mi><mn>2</mn></msub><mi mathvariant="normal">∣</mi></mrow></mfrac><mo>+</mo><mfrac><mn>1</mn><mrow><mi mathvariant="normal">∣</mi><mi>P</mi><msub><mi>F</mi><mn>2</mn></msub><mi mathvariant="normal">∣</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><mn>1</mn><mo>+</mo><msub><mi>λ</mi><mn>2</mn></msub></mrow><mrow><mi mathvariant="normal">∣</mi><mi>P</mi><msub><mi>F</mi><mn>2</mn></msub><mi mathvariant="normal">∣</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><mn>2</mn><mi>a</mi></mrow><msup><mi>b</mi><mn>2</mn></msup></mfrac><mo>=</mo><msqrt><mn>6</mn></msqrt><mspace linebreak="newline"></mspace><mo>∴</mo><msqrt><mn>6</mn></msqrt><mo stretchy="false">(</mo><mi mathvariant="normal">∣</mi><mi>P</mi><msub><mi>F</mi><mn>1</mn></msub><mi mathvariant="normal">∣</mi><mo>+</mo><mi mathvariant="normal">∣</mi><mi>P</mi><msub><mi>F</mi><mn>2</mn></msub><mi mathvariant="normal">∣</mi><mo stretchy="false">)</mo><mo>=</mo><mn>2</mn><mo>+</mo><msub><mi>λ</mi><mn>1</mn></msub><mo>+</mo><msub><mi>λ</mi><mn>2</mn></msub><mo>=</mo><mn>2</mn><mo>×</mo><mn>6</mn><mo>=</mo><mn>12</mn><mspace linebreak="newline"></mspace><mo>∴</mo><msub><mi>λ</mi><mn>1</mn></msub><mo>+</mo><msub><mi>λ</mi><mn>2</mn></msub><mo>=</mo><mn>10</mn></mrow><annotation encoding="application/x-tex">\because\frac{1}{|AF_1|}+\frac{1}{|PF_1|}=\frac{1+\lambda_1}{|PF_1|}=\frac{2a}{b^2}=\sqrt{6},\frac{1}{|AF_2|}+\frac{1}{|PF_2|}=\frac{1+\lambda_2}{|PF_2|}=\frac{2a}{b^2}=\sqrt{6}\\\therefore\sqrt{6}(|PF_1|+|PF_2|)=2+\lambda_1+\lambda_2=2\times6=12\\\therefore\lambda_1+\lambda_2=10</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.69224em;vertical-align:0em;"></span><span class="mrel amsrm">∵</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:2.25744em;vertical-align:-0.936em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.32144em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">∣</span><span class="mord mathnormal">A</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord">∣</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.936em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:2.25744em;vertical-align:-0.936em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.32144em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">∣</span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord">∣</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.936em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:2.30744em;vertical-align:-0.936em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.37144em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">∣</span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord">∣</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord"><span class="mord mathnormal">λ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.936em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:2.00744em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.32144em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">b</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.740108em;"><span style="top:-2.9890000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span><span class="mord mathnormal">a</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:2.25744em;vertical-align:-0.936em;"></span><span class="mord sqrt"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.956095em;"><span class="svg-align" style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord" style="padding-left:0.833em;"><span class="mord">6</span></span></span><span style="top:-2.916095em;"><span class="pstrut" style="height:3em;"></span><span class="hide-tail" style="min-width:0.853em;height:1.08em;"><svg width='400em' height='1.08em' viewBox='0 0 400000 1080' preserveAspectRatio='xMinYMin slice'><path d='M95,702c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z'/></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.08390500000000001em;"><span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.32144em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">∣</span><span class="mord mathnormal">A</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord">∣</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.936em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:2.25744em;vertical-align:-0.936em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.32144em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">∣</span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord">∣</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.936em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:2.30744em;vertical-align:-0.936em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.37144em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">∣</span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord">∣</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord"><span class="mord mathnormal">λ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.936em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:2.00744em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.32144em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">b</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.740108em;"><span style="top:-2.9890000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span><span class="mord mathnormal">a</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1.04em;vertical-align:-0.08390500000000001em;"></span><span class="mord sqrt"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.956095em;"><span class="svg-align" style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord" style="padding-left:0.833em;"><span class="mord">6</span></span></span><span style="top:-2.916095em;"><span class="pstrut" style="height:3em;"></span><span class="hide-tail" style="min-width:0.853em;height:1.08em;"><svg width='400em' height='1.08em' viewBox='0 0 400000 1080' preserveAspectRatio='xMinYMin slice'><path d='M95,702c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z'/></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.08390500000000001em;"><span></span></span></span></span></span></span><span class="mspace newline"></span><span class="base"><span class="strut" style="height:0.69224em;vertical-align:0em;"></span><span class="mrel amsrm">∴</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1.206095em;vertical-align:-0.25em;"></span><span class="mord sqrt"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.956095em;"><span class="svg-align" style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord" style="padding-left:0.833em;"><span class="mord">6</span></span></span><span style="top:-2.916095em;"><span class="pstrut" style="height:3em;"></span><span class="hide-tail" style="min-width:0.853em;height:1.08em;"><svg width='400em' height='1.08em' viewBox='0 0 400000 1080' preserveAspectRatio='xMinYMin slice'><path d='M95,702c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z'/></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.08390500000000001em;"><span></span></span></span></span></span><span class="mopen">(</span><span class="mord">∣</span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord">∣</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">∣</span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord">∣</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.72777em;vertical-align:-0.08333em;"></span><span class="mord">2</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.84444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">λ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.84444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">λ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.72777em;vertical-align:-0.08333em;"></span><span class="mord">2</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">6</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">1</span><span class="mord">2</span></span><span class="mspace newline"></span><span class="base"><span class="strut" style="height:0.69224em;vertical-align:0em;"></span><span class="mrel amsrm">∴</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.84444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">λ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.84444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">λ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">1</span><span class="mord">0</span></span></span></span></span></p><p>几乎是“秒杀”，那么我们来看一道高考题。</p><p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mtext>已知椭圆</mtext><mfrac><msup><mi>x</mi><mn>2</mn></msup><mn>3</mn></mfrac><mo>+</mo><mfrac><msup><mi>y</mi><mn>2</mn></msup><mn>2</mn></mfrac><mo>=</mo><mn>1</mn><mtext>的左右焦点分别为</mtext><msub><mi>F</mi><mn>1</mn></msub><mtext>，</mtext><msub><mi>F</mi><mn>2</mn></msub><mtext>,过点</mtext><msub><mi>F</mi><mn>1</mn></msub><mtext>的直线交椭圆于B,D两点，过点</mtext><mspace linebreak="newline"></mspace><msub><mi>F</mi><mn>2</mn></msub><mtext>的直线交椭圆于A,C两点，且</mtext><mi>A</mi><mi>C</mi><mi mathvariant="normal">⊥</mi><mi>B</mi><mi>D</mi><mo separator="true">,</mo><mtext>垂足为P,求四边形ABCD的面积最小值。</mtext></mrow><annotation encoding="application/x-tex">\text{已知椭圆}\frac{x^2}{3}+\frac{y^2}{2}=1\text{的左右焦点分别为}F_1，F_2\text{,过点}F_1\text{的直线交椭圆于B,D两点，过点}\\F_2\text{的直线交椭圆于A,C两点，且}AC\bot BD,\text{垂足为P,求四边形ABCD的面积最小值。}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.177108em;vertical-align:-0.686em;"></span><span class="mord text"><span class="mord cjk_fallback">已知椭圆</span></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.491108em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">3</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:2.177108em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.491108em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.8777699999999999em;vertical-align:-0.19444em;"></span><span class="mord">1</span><span class="mord text"><span class="mord cjk_fallback">的左右焦点分别为</span></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord cjk_fallback">，</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord text"><span class="mord">,</span><span class="mord cjk_fallback">过点</span></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord text"><span class="mord cjk_fallback">的直线交椭圆于</span><span class="mord">B,D</span><span class="mord cjk_fallback">两点，过点</span></span></span><span class="mspace newline"></span><span class="base"><span class="strut" style="height:0.8888799999999999em;vertical-align:-0.19444em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord text"><span class="mord cjk_fallback">的直线交椭圆于</span><span class="mord">A,C</span><span class="mord cjk_fallback">两点，且</span></span><span class="mord mathnormal">A</span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="mord">⊥</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord text"><span class="mord cjk_fallback">垂足为</span><span class="mord">P,</span><span class="mord cjk_fallback">求四边形</span><span class="mord">ABCD</span><span class="mord cjk_fallback">的面积最小值。</span></span></span></span></span></span></p><p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mtext>设AC,BD的倾斜角分别为</mtext><mi>α</mi><mo separator="true">,</mo><mi>β</mi><mo separator="true">,</mo><mtext>看到垂直的条件，我们不妨设</mtext><mi>α</mi><mo>=</mo><mi>β</mi><mo>+</mo><mfrac><mi>π</mi><mn>2</mn></mfrac><mspace linebreak="newline"></mspace><mtext>那么</mtext><mi>cos</mi><mo>⁡</mo><mi>α</mi><mo>=</mo><mo>−</mo><mi>sin</mi><mo>⁡</mo><mi>β</mi><mspace linebreak="newline"></mspace><mtext>所以</mtext><mi>S</mi><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mi mathvariant="normal">∣</mi><mi>A</mi><mi>C</mi><mi mathvariant="normal">∣</mi><mi mathvariant="normal">∣</mi><mi>B</mi><mi>D</mi><mi mathvariant="normal">∣</mi><mo>=</mo><mfrac><mrow><mn>2</mn><msup><mi>a</mi><mn>2</mn></msup><msup><mi>b</mi><mn>4</mn></msup></mrow><mrow><mo stretchy="false">(</mo><msup><mi>a</mi><mn>2</mn></msup><mo>−</mo><msup><mi>c</mi><mn>2</mn></msup><msup><mo><mi>cos</mi><mo>⁡</mo></mo><mn>2</mn></msup><mi>β</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msup><mi>a</mi><mn>2</mn></msup><mo>−</mo><msup><mi>c</mi><mn>2</mn></msup><msup><mo><mi>sin</mi><mo>⁡</mo></mo><mn>2</mn></msup><mi>β</mi><mo stretchy="false">)</mo></mrow></mfrac><mspace linebreak="newline"></mspace><mo>=</mo><mfrac><mn>24</mn><mrow><mn>6</mn><mo>+</mo><mfrac><mn>1</mn><mn>4</mn></mfrac><mi>s</mi><mi>i</mi><msup><mi>n</mi><mn>2</mn></msup><mn>2</mn><mi>β</mi></mrow></mfrac><mspace linebreak="newline"></mspace><mo>≤</mo><mfrac><mn>96</mn><mn>25</mn></mfrac></mrow><annotation encoding="application/x-tex">\text{设AC,BD的倾斜角分别为}\alpha,\beta,\text{看到垂直的条件，我们不妨设}\alpha=\beta+\frac{\pi}{2}\\\text{那么}\cos\alpha=-\sin\beta\\\text{所以}S=\frac{1}{2}|AC||BD|=\frac{2a^2b^4}{(a^2-c^2\cos^2\beta)(a^2-c^2\sin^2\beta)}\\=\frac{24}{6+\frac{1}{4}sin^22\beta}\\\leq\frac{96}{25}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8888799999999999em;vertical-align:-0.19444em;"></span><span class="mord text"><span class="mord cjk_fallback">设</span><span class="mord">AC,BD</span><span class="mord cjk_fallback">的倾斜角分别为</span></span><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal" style="margin-right:0.05278em;">β</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord text"><span class="mord cjk_fallback">看到垂直的条件，我们不妨设</span></span><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.8888799999999999em;vertical-align:-0.19444em;"></span><span class="mord mathnormal" style="margin-right:0.05278em;">β</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1.7935600000000003em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.10756em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">π</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span><span class="mspace newline"></span><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord text"><span class="mord cjk_fallback">那么</span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mop">cos</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.8888799999999999em;vertical-align:-0.19444em;"></span><span class="mord">−</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mop">sin</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal" style="margin-right:0.05278em;">β</span></span><span class="mspace newline"></span><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord text"><span class="mord cjk_fallback">所以</span></span><span class="mord mathnormal" style="margin-right:0.05764em;">S</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:2.00744em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.32144em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord">∣</span><span class="mord mathnormal">A</span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="mord">∣</span><span class="mord">∣</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="mord">∣</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:2.5029760000000003em;vertical-align:-1.011868em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.491108em;"><span style="top:-2.2381320000000002em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.740108em;"><span style="top:-2.9890000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord"><span class="mord mathnormal">c</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.740108em;"><span style="top:-2.9890000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mop"><span class="mop">cos</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.740108em;"><span style="top:-2.9890000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal" style="margin-right:0.05278em;">β</span><span class="mclose">)</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.740108em;"><span style="top:-2.9890000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord"><span class="mord mathnormal">c</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.740108em;"><span style="top:-2.9890000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mop"><span class="mop">sin</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.871868em;"><span style="top:-3.12076em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal" style="margin-right:0.05278em;">β</span><span class="mclose">)</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">b</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">4</span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.011868em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span><span class="mspace newline"></span><span class="base"><span class="strut" style="height:0.36687em;vertical-align:0em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:2.401548em;vertical-align:-1.080108em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.32144em;"><span style="top:-2.2648919999999997em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">6</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.845108em;"><span style="top:-2.6550000000000002em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">4</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord mathnormal">s</span><span class="mord mathnormal">i</span><span class="mord"><span class="mord mathnormal">n</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.740108em;"><span style="top:-2.9890000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mord">2</span><span class="mord mathnormal" style="margin-right:0.05278em;">β</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span><span class="mord">4</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.080108em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span><span class="mspace newline"></span><span class="base"><span class="strut" style="height:0.7719400000000001em;vertical-align:-0.13597em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:2.00744em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.32144em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span><span class="mord">5</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">9</span><span class="mord">6</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></p><p>此外利用好点到焦点焦点和点到准线还能处理一些角的问题，如下。</p><p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mtext>已知抛物线</mtext><msup><mi>y</mi><mn>2</mn></msup><mo>=</mo><mn>2</mn><mi>p</mi><mi>x</mi><mo stretchy="false">(</mo><mi>p</mi><mo>&gt;</mo><mn>0</mn><mo stretchy="false">)</mo><mtext>的焦点为F，准线为l，l与x轴交点为M,过点F的直线AB交抛物线于AB两点</mtext><mspace linebreak="newline"></mspace><mi>A</mi><mtext>在x轴上方,</mtext><mi mathvariant="normal">∠</mi><mi>A</mi><mi>M</mi><mi>B</mi><mo>=</mo><mfrac><mi>π</mi><mn>3</mn></mfrac><mo separator="true">,</mo><mi mathvariant="normal">∣</mi><mi>A</mi><mi>F</mi><mi mathvariant="normal">∣</mi><mo>=</mo><msqrt><mn>2</mn></msqrt><mo>+</mo><msqrt><mn>3</mn></msqrt><mtext>，试求抛物线方程.</mtext></mrow><annotation encoding="application/x-tex">\text{已知抛物线}y^2=2px(p&gt;0)\text{的焦点为F，准线为l，l与x轴交点为M,过点F的直线AB交抛物线于AB两点}\\A\text{在x轴上方,}∠AMB=\frac{\pi}{3},|AF|=\sqrt{2}+\sqrt{3}，\text{试求抛物线方程.}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0585479999999998em;vertical-align:-0.19444em;"></span><span class="mord text"><span class="mord cjk_fallback">已知抛物线</span></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641079999999999em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">2</span><span class="mord mathnormal">p</span><span class="mord mathnormal">x</span><span class="mopen">(</span><span class="mord mathnormal">p</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">&gt;</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">0</span><span class="mclose">)</span><span class="mord text"><span class="mord cjk_fallback">的焦点为</span><span class="mord">F</span><span class="mord cjk_fallback">，准线为</span><span class="mord">l</span><span class="mord cjk_fallback">，</span><span class="mord">l</span><span class="mord cjk_fallback">与</span><span class="mord">x</span><span class="mord cjk_fallback">轴交点为</span><span class="mord">M,</span><span class="mord cjk_fallback">过点</span><span class="mord">F</span><span class="mord cjk_fallback">的直线</span><span class="mord">AB</span><span class="mord cjk_fallback">交抛物线于</span><span class="mord">AB</span><span class="mord cjk_fallback">两点</span></span></span><span class="mspace newline"></span><span class="base"><span class="strut" style="height:0.8866799999999999em;vertical-align:-0.19444em;"></span><span class="mord mathnormal">A</span><span class="mord text"><span class="mord cjk_fallback">在</span><span class="mord">x</span><span class="mord cjk_fallback">轴上方</span><span class="mord">,</span></span><span class="mord">∠</span><span class="mord mathnormal">A</span><span class="mord mathnormal" style="margin-right:0.10903em;">M</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1.7935600000000003em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.10756em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">3</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">π</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">∣</span><span class="mord mathnormal">A</span><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="mord">∣</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1.04em;vertical-align:-0.08390500000000001em;"></span><span class="mord sqrt"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.956095em;"><span class="svg-align" style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord" style="padding-left:0.833em;"><span class="mord">2</span></span></span><span style="top:-2.916095em;"><span class="pstrut" style="height:3em;"></span><span class="hide-tail" style="min-width:0.853em;height:1.08em;"><svg width='400em' height='1.08em' viewBox='0 0 400000 1080' preserveAspectRatio='xMinYMin slice'><path d='M95,702c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z'/></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.08390500000000001em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1.04em;vertical-align:-0.08390500000000001em;"></span><span class="mord sqrt"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.956095em;"><span class="svg-align" style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord" style="padding-left:0.833em;"><span class="mord">3</span></span></span><span style="top:-2.916095em;"><span class="pstrut" style="height:3em;"></span><span class="hide-tail" style="min-width:0.853em;height:1.08em;"><svg width='400em' height='1.08em' viewBox='0 0 400000 1080' preserveAspectRatio='xMinYMin slice'><path d='M95,702c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z'/></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.08390500000000001em;"><span></span></span></span></span></span><span class="mord cjk_fallback">，</span><span class="mord text"><span class="mord cjk_fallback">试求抛物线方程</span><span class="mord">.</span></span></span></span></span></span></p><p><img src="https://cdn.jsdelivr.net/gh/Yurchiu/PicGo/fa46f26bbae76cecd00255dc64a2a596.png" alt="" /></p><p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mtext>如图设</mtext><mi mathvariant="normal">∠</mi><mi>A</mi><mi>M</mi><mi>F</mi><mo>=</mo><mi>φ</mi><mo separator="true">,</mo><mtext>直线AB的倾斜角为</mtext><mi>β</mi><mo separator="true">,</mo><mtext>则有</mtext><mi>sin</mi><mo>⁡</mo><mi>β</mi><mo>=</mo><mfrac><mrow><mi>C</mi><mi>M</mi></mrow><mrow><mi>A</mi><mi>F</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><mi>C</mi><mi>M</mi></mrow><mrow><mi>A</mi><mi>C</mi></mrow></mfrac><mo>=</mo><mi>tan</mi><mo>⁡</mo><mi>φ</mi><mspace linebreak="newline"></mspace><mtext>同理</mtext><mi>sin</mi><mo>⁡</mo><mi>β</mi><mo>=</mo><mi>tan</mi><mo>⁡</mo><mi mathvariant="normal">∠</mi><mi>B</mi><mi>M</mi><mi>F</mi><mo separator="true">,</mo><mtext>所以</mtext><mi mathvariant="normal">∠</mi><mi>A</mi><mi>M</mi><mi>F</mi><mo>=</mo><mi mathvariant="normal">∠</mi><mi>B</mi><mi>M</mi><mi>F</mi><mspace linebreak="newline"></mspace><mo>∴</mo><mi>sin</mi><mo>⁡</mo><mi>β</mi><mo>=</mo><mfrac><msqrt><mn>3</mn></msqrt><mn>3</mn></mfrac><mo separator="true">,</mo><mi mathvariant="normal">∣</mi><mi>A</mi><mi>F</mi><mi mathvariant="normal">∣</mi><mo>=</mo><mfrac><mi>p</mi><mrow><mn>1</mn><mo>−</mo><mi>cos</mi><mo>⁡</mo><mi>β</mi></mrow></mfrac><mo>=</mo><msqrt><mn>2</mn></msqrt><mo>+</mo><msqrt><mn>3</mn></msqrt><mspace linebreak="newline"></mspace><mtext>解得：</mtext><mi>p</mi><mo>=</mo><mfrac><msqrt><mn>3</mn></msqrt><mn>3</mn></mfrac><mo separator="true">,</mo><mtext>所以抛物线方程为：</mtext><msup><mi>y</mi><mn>2</mn></msup><mo>=</mo><mfrac><mrow><mn>2</mn><msqrt><mn>3</mn></msqrt></mrow><mn>3</mn></mfrac><mi>x</mi></mrow><annotation encoding="application/x-tex">\text{如图设}\angle AMF=\varphi,\text{直线AB的倾斜角为}\beta,\text{则有}\sin\beta=\frac{CM}{AF}=\frac{CM}{AC}=\tan\varphi\\\text{同理}\sin\beta=\tan\angle BMF,\text{所以}\angle AMF=\angle BMF\\\therefore\sin\beta=\frac{\sqrt{3}}{3},|AF|=\frac{p}{1-\cos\beta}=\sqrt{2}+\sqrt{3}\\\text{解得：}p=\frac{\sqrt{3}}{3},\text{所以抛物线方程为：}y^2=\frac{2\sqrt{3}}{3}x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.69224em;vertical-align:0em;"></span><span class="mord text"><span class="mord cjk_fallback">如图设</span></span><span class="mord">∠</span><span class="mord mathnormal">A</span><span class="mord mathnormal" style="margin-right:0.10903em;">M</span><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.8888799999999999em;vertical-align:-0.19444em;"></span><span class="mord mathnormal">φ</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord text"><span class="mord cjk_fallback">直线</span><span class="mord">AB</span><span class="mord cjk_fallback">的倾斜角为</span></span><span class="mord mathnormal" style="margin-right:0.05278em;">β</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord text"><span class="mord cjk_fallback">则有</span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mop">sin</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal" style="margin-right:0.05278em;">β</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:2.04633em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.36033em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">A</span><span class="mord mathnormal" style="margin-right:0.13889em;">F</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="mord mathnormal" style="margin-right:0.10903em;">M</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:2.04633em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.36033em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">A</span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="mord mathnormal" style="margin-right:0.10903em;">M</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.80952em;vertical-align:-0.19444em;"></span><span class="mop">tan</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal">φ</span></span><span class="mspace newline"></span><span class="base"><span class="strut" style="height:0.8888799999999999em;vertical-align:-0.19444em;"></span><span class="mord text"><span class="mord cjk_fallback">同理</span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mop">sin</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal" style="margin-right:0.05278em;">β</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.8866799999999999em;vertical-align:-0.19444em;"></span><span class="mop">tan</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">∠</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mord mathnormal" style="margin-right:0.10903em;">M</span><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord text"><span class="mord cjk_fallback">所以</span></span><span class="mord">∠</span><span class="mord mathnormal">A</span><span class="mord mathnormal" style="margin-right:0.10903em;">M</span><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.69224em;vertical-align:0em;"></span><span class="mord">∠</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mord mathnormal" style="margin-right:0.10903em;">M</span><span class="mord mathnormal" style="margin-right:0.13889em;">F</span></span><span class="mspace newline"></span><span class="base"><span class="strut" style="height:0.69224em;vertical-align:0em;"></span><span class="mrel amsrm">∴</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.8888799999999999em;vertical-align:-0.19444em;"></span><span class="mop">sin</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal" style="margin-right:0.05278em;">β</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:2.27022em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.5842200000000002em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">3</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord sqrt"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.90722em;"><span class="svg-align" style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord" style="padding-left:0.833em;"><span class="mord">3</span></span></span><span style="top:-2.86722em;"><span class="pstrut" style="height:3em;"></span><span class="hide-tail" style="min-width:0.853em;height:1.08em;"><svg width='400em' height='1.08em' viewBox='0 0 400000 1080' preserveAspectRatio='xMinYMin slice'><path d='M95,702c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z'/></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.13278em;"><span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">∣</span><span class="mord mathnormal">A</span><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="mord">∣</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1.988em;vertical-align:-0.8804400000000001em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.1075599999999999em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mop">cos</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal" style="margin-right:0.05278em;">β</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">p</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.8804400000000001em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1.04em;vertical-align:-0.08390500000000001em;"></span><span class="mord sqrt"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.956095em;"><span class="svg-align" style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord" style="padding-left:0.833em;"><span class="mord">2</span></span></span><span style="top:-2.916095em;"><span class="pstrut" style="height:3em;"></span><span class="hide-tail" style="min-width:0.853em;height:1.08em;"><svg width='400em' height='1.08em' viewBox='0 0 400000 1080' preserveAspectRatio='xMinYMin slice'><path d='M95,702c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z'/></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.08390500000000001em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1.04em;vertical-align:-0.08390500000000001em;"></span><span class="mord sqrt"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.956095em;"><span class="svg-align" style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord" style="padding-left:0.833em;"><span class="mord">3</span></span></span><span style="top:-2.916095em;"><span class="pstrut" style="height:3em;"></span><span class="hide-tail" style="min-width:0.853em;height:1.08em;"><svg width='400em' height='1.08em' viewBox='0 0 400000 1080' preserveAspectRatio='xMinYMin slice'><path d='M95,702c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z'/></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.08390500000000001em;"><span></span></span></span></span></span></span><span class="mspace newline"></span><span class="base"><span class="strut" style="height:0.8777699999999999em;vertical-align:-0.19444em;"></span><span class="mord text"><span class="mord cjk_fallback">解得：</span></span><span class="mord mathnormal">p</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:2.27022em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.5842200000000002em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">3</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord sqrt"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.90722em;"><span class="svg-align" style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord" style="padding-left:0.833em;"><span class="mord">3</span></span></span><span style="top:-2.86722em;"><span class="pstrut" style="height:3em;"></span><span class="hide-tail" style="min-width:0.853em;height:1.08em;"><svg width='400em' height='1.08em' viewBox='0 0 400000 1080' preserveAspectRatio='xMinYMin slice'><path d='M95,702c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z'/></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.13278em;"><span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord text"><span class="mord cjk_fallback">所以抛物线方程为：</span></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641079999999999em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:2.27022em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.5842200000000002em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">3</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span><span class="mord sqrt"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.90722em;"><span class="svg-align" style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord" style="padding-left:0.833em;"><span class="mord">3</span></span></span><span style="top:-2.86722em;"><span class="pstrut" style="height:3em;"></span><span class="hide-tail" style="min-width:0.853em;height:1.08em;"><svg width='400em' height='1.08em' viewBox='0 0 400000 1080' preserveAspectRatio='xMinYMin slice'><path d='M95,702c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z'/></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.13278em;"><span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord mathnormal">x</span></span></span></span></span></p><p>同样椭圆双曲线也有类似结论，证明也很简单，大家不妨自行尝试推导。</p><p>在圆锥曲线部分，有关焦点的问题可以说是俯拾即是，利用上述结论可以迅速解决大部分这类问题，节省大量时间，但同时不要忘记对于基础联立韦达的解法对一般情况进行求解，否则遇到非焦点问题可就是“提笔四顾心茫然”，感谢您的阅读，希望这篇文章能够帮助到你。</p>]]></content>
      
      
      <categories>
          
          <category> 数学 </category>
          
      </categories>
      
      
        <tags>
            
            <tag> 数学 </tag>
            
        </tags>
      
    </entry>
    
    
    
    <entry>
      <title>《幽默报 11》</title>
      <link href="/2024/08/02/%E3%80%8A%E5%B9%BD%E9%BB%98%E6%8A%A5%2011%E3%80%8B/"/>
      <url>/2024/08/02/%E3%80%8A%E5%B9%BD%E9%BB%98%E6%8A%A5%2011%E3%80%8B/</url>
      
        <content type="html"><![CDATA[<p>这玩意出到第十一期，在座的各位都有责任。往期幽默报：<a href="https://yz-hs.github.io/categories/%E6%B0%B4/%E5%B9%BD%E9%BB%98%E6%8A%A5/">https://yz-hs.github.io/categories/水/幽默报/</a>。</p><span id="more"></span><h1 id="1"><a class="markdownIt-Anchor" href="#1"></a> 1</h1><p>激进派和保守派，激进派语言和行动都很激进，保守派认为激进派太保守了。</p><p>蟹老板：你被开除了<br />海绵宝宝：蟹老板<br />蟹老板：不用谢[doge]</p><p>这是一种 积 极的创新态度<br />这是一种 你 不会懂的热爱<br />这是一种 太 过超前的艺术<br />这是一种 美 丽的人生意境</p><p>这可不是什么红石，这种东西叫做蓝莓，俗称苹果。生长在撒哈拉沙漠的雨林地带。因外形酷似企鹅，所以我们又喜欢叫他北极熊。你们这些人，连仙人掌都不知道，就不要乱说他是西瓜好吗？再说一遍这个橘子叫做猕猴桃</p><p>刚才去吃火锅，看见火锅店的墙上提醒，羊是自己养的，菜是自己种的油是自己榨的，提醒顾客放心使用。买单的时候我，悄悄告诉老板，钱是我自己印的，请放心使用，老板追了我好几条街没追上，真有意思，腿是我自己长的，想往哪里跑就往哪里跑</p><p>刚才去吃火锅，看见【火锅店老板】操纵提线，让它们作响：“我向【自己养的羊】祈求，向【自己种的菜】祈求，向【自己榨的油】祈求，怎么不推荐我的【三合一套餐】！只需要你那小小的【买单】！”买单的时候我，悄悄告诉老板：“我向【自己印的钱】祈求，多拿点！”，老板追了像我这样的【史莱姆】好几条【女王豪宅】走廊没追上，这个实验真有意思，老板向【皮皮斯】祈求，想往哪里扔就往哪里仍。【光之苠】，如果你下滑如【流水】，那我祝你现在有机会返回1997成为【【大人物】】！</p><p>□□□□，□□《□□》□□□□□□□□□□□□□□□□□□□□□□。□□□□□□□□□□「□□□」□□□□□，□□□，□□□□□□□□□□「□□□」，□□□□□□。□□□□□□□□「□□□」□□□□□，□□□□□□□□□□□□□、□□□□□□□□，□□□□□□□□□，□□□□□□□——□□，□□□□「□□」□□□。</p><p>正常人做饭： <code>(没放盐,放多了)</code><br />初学者做饭： <code>[没放盐,放多了]</code><br />我做饭： <code>&#123;没放盐,放多了&#125;</code></p><h1 id="2"><a class="markdownIt-Anchor" href="#2"></a> 2</h1><p>1、曾经有教育家做了一个实验，给中国孩子和美国孩子一杯水，让他们不用火就让水沸腾起来。中国孩子拿水在太阳下晒了一天，没有如愿。而聪明的美国孩子拿来四杯25度的水混合到一起，轻松地把水温升到了沸腾的100度。僵化的思维，落后的体制永远无法培养有创新意识的大科学家。</p><p>2、曾经有教育家做了一个实验，给中国孩子和美国孩子一杯水，让他们不用火就让水沸腾起来。中国孩子拿来四杯25度的水混合到一起，轻松地把水温升到了沸腾的100度。而朴实的美国孩子拿着水在太阳下晒着，静静等待，最终水也升到了100度。虽然美国孩子输了，但是虽败犹荣，因为美国教育下的孩子诚实而有品质，从来不会耍小聪明。只有这样的孩子，以后才能成为正直的人。</p><p>3、 曾经有教育家做了一个实验，给一群中国孩子和一群美国孩子一人一杯水，让他们不用火就让水沸腾起来。中国孩子都把水放在太阳底下晒了一天也没到100度，而聪明的美国孩子们在一起经过商讨以后，把水混在了一块，刚好四杯25度的水混合成了100度。僵化的思维，落后的体制永远无法培养有创新意识的大科学家，也永远无法培养出有合作意识、互助精神的青年。</p><p>4、曾经有教育家做了一个实验，给一群中国孩子和一群美国孩子一人一杯水，让他们不用火就让水沸腾起来。中国孩子在一起经过商讨以后，把水混在了一块，刚好四杯25度的水混合成了100度。而美国孩子都把水放在太阳底下晒了一天，终于把水温升到了100度。中国的孩子永远学不会自己解决问题，而总是想依赖群体的力量，而美国孩子能够独立自主，自力更生，这样的民族才有希望。</p><p>5、 曾经有教育家做了一个实验，给中国孩子和美国孩子一杯水，让他们不用火就让水沸腾起来。中国孩子却在太阳下把水晒了一整天，非要把水温升到100度不可，不然就不回家吃饭。最终还是失败了。美国孩子从小就懂得知足与放弃，而中国的孩子在填鸭教育的影响下，依旧愚蠢地执着坚持不可能的事情。这样永远也培养不出大科学家。</p><p>6、曾经有教育家做了一个实验，给中国孩子和美国孩子一杯水，让他们不用火就让水沸腾起来。中国孩子直接认输了。而美国孩子偷来他爸爸的勃朗宁 一下子把教育家毙了，从此再也没有被教育家研究的烦恼了。从小培养反抗强权的意识，比任何教育都重要。</p><h1 id="3"><a class="markdownIt-Anchor" href="#3"></a> 3</h1><p>一位少女在旅途中意外发现了平行时空中基于无生源论诞生的另一个自己，短暂的初次见面后两人的分离使少女踏上了寻找自我和真相的旅途，重重苦寻和错会后两人终于再次相遇并发现了少女诞生的真相，两位少女在星空下许诺要永远陪伴对方不离不弃，少女的身世最终或许只有她“一人”知晓……</p><p>在废墟之中，我遇见了她。</p><p>四周毫无生机 ，锈蚀的气息弥漫着，管道拐角处的水滴声是我唯一听见的声音。</p><p>我漫无目的地摸索。</p><p>在巨大的结构建筑中央，我发现一个若隐若现的身影。</p><p>水滴仍然不停歇地落下，宛如淅淅沥沥的春雨，给这终焉的世界增添了少许的生机。</p><p>在她的叙述中，我看见了无数被珍藏的记忆。</p><p>人们相遇，相知，再到别离。</p><p>无论是美好抑或悲伤，都是世界曾经的记忆与生命。</p><p>“或许我们可以这样一直走下去。”</p><p>“新生的世界，在此萌芽。”</p><h1 id="4"><a class="markdownIt-Anchor" href="#4"></a> 4</h1><p>那些想要自杀而犹豫不决的人，其实是在经受着一场地狱之火的烧灼。</p><p>人的本能就是生存，要么为何害怕高高的楼层呢？</p><p>凡事都需要动机，比如利益，比如快感，比如精神失常。</p><p>从自杀的念头，到自杀的事实，是一个动机积累和抗衡的过程。比如，对自己的经济状况、家庭、性别、地位不满等，可以作为自杀的理由。家人的牵挂、生活的不舍等，可以作为活下来的理由。它们积累，抗衡，把你来回拉拽。很痛苦，不是吗？</p><p>当你走向窗台，低头看着或车水马龙，或平静宁和的世界，恐惧就侵占了你的内心。所谓恐高症，其实来源于人类的本能。这是你的大敌，因为是它阻止了你自杀，让你卡在鬼门关，让你受地狱之火的烧灼。这是活下来的最大的理由，没有之一。</p><p>于是，你拼命寻找自杀的理由。药物滥用，摧毁自己的身体；乱交，败坏自己的人格；反社会行为，放弃自己的意义。</p><p>哦，不，你还是打败不了那生存的本能。那窗台已经被踏上了无数次，见证了你的无数次煎熬。似乎，那窗台是专门给你准备的历练之地。</p><p>你想，为什么要毁掉自己呢？我受不了一心想死却不敢了结自己所带来的煎熬。这就是那地狱之火。那并非常人之可胜受，它让你浑浑噩噩，让你乱麻一团，让你痛苦万分。为什么要毁掉自己呢？因为自杀的念头已然驱之不去，活下去的欲望与之共存。一山不容二虎，二虎会将你折磨至疯狂。助长吧，自杀吧。</p><p>一天晚上，你又来到窗台，依旧下不去决心。希望一阵风把你刮下高楼的念头，想必是有的吧。如果那样如果自杀成功，有什么意义呢？</p><p>呵呵，人类，真是精妙啊。</p><p>那些在窗台的人，正在接受地狱之火的烧灼。无论过后决定是生是死，他们都是勇士，绝非懦夫。</p><p>没人想悄无声息地死。那些寻死的人，其实是为了摆脱痛苦，否则他们也不会抗争人类的本能。他们想要留下存在的痕迹。遗书，聊天记录，事迹……<br />他们是要延缓第二次死亡和第三次死亡的来临。“死而不亡者寿”，就是这个道理吧。</p><p>所以，自杀又何尝不等同于寻求生存呢？</p><p>深夜发病，观点激进，勿在意。</p><h1 id="5"><a class="markdownIt-Anchor" href="#5"></a> 5</h1><p>分享一下治疗玉玉症的中药方子，个人总结出来的，非常有效，谨供参考</p><p>周一去麦邓劳中医馆，老字号中医馆，也有很多网友推荐，不多赘述<br />推荐<br />灀棘治湿汗煲150克，红篼槟80克，盅枢调110克，必要时辅以麦楦枫一剂</p><p>周二或者周三去打美乐中医理疗中心，这家是专门做脾飒的，属于是后起之秀，个人认为品质优于弼圣窠<br />推荐<br />孛茛多多肉橡粉松脾飒 12寸<br />喜瓣牙风晴香苌肉姜脾飒 12寸<br />辅以菠萝蒎2皿</p><p>周四去恳悳基中药房，这家的药材品质好是好，但是有些良方只能星期四拿到，每次去都人满为患，建议提前预定<br />推荐<br />蒲柿淡榙 6支<br />热蜡襄古棘 15支</p><h1 id="6"><a class="markdownIt-Anchor" href="#6"></a> 6</h1><p>你至少需要先填写《<strong>关于修改生死簿的申请表</strong>》、《<strong>关于修改生死簿的情况说明</strong>》、《<strong>关于修改生死簿具体问题的报告</strong>》，提交天庭秘书处并耐心等待10个工作日，收到一份《<strong>关于你府申请修改生死簿问题的批复</strong>》。</p><p>《批复》中明确要求：</p><p>一、高度重视此次生死簿修改问题；</p><p>二、统一思想、抓好落实，切实严把生死簿修改关，通过修改生死簿提高地府业务工作水平；</p><p>三、紧跟天庭党委工作指导思想，把握仙界年度发展蓬勃势头，把修改生死簿工作提高到仙界整体工作落实的高度上来；</p><p>四、落实责任、具体到人，对修改生死簿中各方面具体责任要落实到位；</p><p>五、整理汇总、请示汇报，在修改生死簿工作推进的全环节中，要与天庭党委随时保持沟通；</p><p>六、数字赋能、提高效率，重视数字化管理工作在生死簿修改中的作用，提高生死簿修改工作效能。</p><p>地府在接到《批复》后，要及时召开《<strong>关于修改生死簿工作任务部署会</strong>》，召集五方鬼帝、十殿阎罗、六案功曹、六司主事出席，东岳大帝、地藏王菩萨列席，各鬼差参加。</p><p><img src="https://pic1.zhimg.com/80/v2-28ec981e74851477be21b116c7e7969d_720w.webp?source=2c26e567" alt="" /></p><p><img src="https://picx.zhimg.com/80/v2-9657e88c97a985817a304da215b9c14a_720w.webp?source=2c26e567" alt="" /></p><p>会议在雄壮的《哀乐》中开幕。</p><p>首先由东方鬼帝主持会议并宣布会议议程，会议一共有八项议程：</p><p>一、奏《哀乐》；</p><p>二、由南方鬼帝宣读天庭《关于你府申请修改生死簿问题的批复》；</p><p>三、由西方鬼帝就《批复》精神进行宣讲教育；</p><p>四、由中央鬼帝安排部署生死簿修改工作；</p><p>五、由北方鬼帝对生死簿修改工作中的行政工作作指示；</p><p>六、由东方鬼帝对生死簿修改工作中的后勤保障工作作指示；</p><p>七、请酆都大帝作重要指示；</p><p>八、奏《葬礼进行曲》。</p><p>会后，根据酆都大帝重要指示精神，各部门开始就生死簿修改工作开始进行具体动作。</p><p>首先是中央鬼帝府根据会议精神下发《关于修改生死簿问题的通知》，通知下发到六案功曹、六司主事一级，通知中明确了本次生死簿修改工作的指导精神、安排部署、行动步骤、几点意见。</p><p>各司接到通知后，要按照通知精神拟定工作计划，按照时间节点对生死簿修改工作进行分工，并上报中央鬼帝府，抄送各鬼帝府备案。</p><p>为切实落实天庭党委年初《<strong>关于切实反对官僚主义、消除“文山会海”的通知</strong>》精神，各工作环节还必须要认真梳理总结，不走形式，避免过多占用日常工作时间，因此大家要充分利用下班时间补充、补写工作报告，填写各类表格，搞好调查走访，收集群众意见，以上内容全部以表格形式呈现，避免“八股文”、“四六句”。</p><p>最后由中央鬼帝府形成《<strong>关于修改生死簿问题的调查报告</strong>》、《<strong>关于修改生死簿问题的处理意见</strong>》，《报告》、《意见》中认为：</p><p>一、本次事件的主要责任人是地府查察司拘魂使黑白无常，主要责任是工作作风简单粗暴，对于修炼有成的孙悟空工作中过于死板、教条，不能灵活掌握生死簿的执行，是导致生死簿被非法修改的直接原因；</p><p>二、地府领导集体对生死簿修改负有领导责任，主要责任是对部属失管失控，不重视思想教育和行政能力培养，导致个别地府工作人员工作作风简单粗暴，灵活性不足，错误拘押修炼有成的孙悟空并导致生死簿被非法修改；</p><p>三、痛定思痛、加强整改，首先在地府中掀起“从严执法、灵活机动、请示汇报”的三项作风整顿行动，广泛教育地府工作人员在日常执法行政中切实落实天庭党委决议，紧跟天庭工作步伐，解决地府工作中长期存在的问题、矛盾；</p><p>四、建议给予具体责任人黑白无常记过处分一次，纪委诫勉谈话；</p><p>五、建议给予地府领导集体警告处分一次；</p><p>六、建议重新修订生死簿，划除孙悟空条目。</p><p>天庭接到《报告》、《意见》之后，呈玉帝审阅通过并在天庭行政工作会上进行了讨论形成决议，基本认可地府《意见》，对《报告》中的情况也给予了充分肯定，并指出三个问题：</p><p>一是地府工作力度偏软、偏散，要加强智慧赋能、数据共享，对于天庭掌握的修炼数据不能及时共享是造成本次事件的深层次原因；</p><p>二是地府领导集体不够团结统一，工作中通气不足、存在隔阂；</p><p>三是地府在工作人员外出作业期间存在跟踪问效不够的问题。</p><p>会议基本同意了地府的处理意见。</p><p>地府在接到《<strong>天庭关于地府修改生死簿问题的会议决议</strong>》之后，认真学习讨论了《决议》并拟定了整改计划，搞了整改“回头看”，并由酆都大帝在天庭党委扩大会上进行了深刻检查。</p><p>最后，一个阴律司办事员小鬼在生死簿上轻轻地勾了一下，“孙悟空”这个条目不复存在。</p><p>酆都大帝独自一人在自己屋子里的时候咬牙切齿的说：</p><p>“改，改，改，改你老木啊改！谁特么再改谁就是孙子！”</p>]]></content>
      
      
      <categories>
          
          <category> 幽默报 </category>
          
      </categories>
      
      
        <tags>
            
            <tag> 幽默报 </tag>
            
        </tags>
      
    </entry>
    
    
    
    <entry>
      <title>LaTeX-Test</title>
      <link href="/2024/07/29/LaTeX-Test/"/>
      <url>/2024/07/29/LaTeX-Test/</url>
      
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68eb06a1884077a7f816370ebeee150f8ea7c73429d26ac9aed91ba1bd8ea0308c3f2229d8e1f38286d0ab5d550787ed5949fb407dc81fe515fef5169ac252e0f0c21946e2cbe1beafc326cd91536fd33c78203abc0c57ee4b7b026f3bcc84fe578a03fc5ffb5cd66f6df51ba0744eaa686c1455b21f123d85829af1b2a6322a5d9f0658f369c4c6f18528440381ad64bf51c107f693d7afc9dd113a5c70e9ebc064895cf9105f01b704ac3600cd1b25cb1c02cdb1997750d85c6ed0c194972e3092cf862da5abd67fe785ba5470054ddc976e9483616a8e0d9c874f8d0faf4f87a8fef48c642d32837d64b5ab9</script>  <div class="hbe hbe-content">    <div class="hbe hbe-input hbe-input-default">      <input class="hbe hbe-input-field hbe-input-field-default" type="password" id="hbePass">      <label class="hbe hbe-input-label hbe-input-label-default" for="hbePass">        <span class="hbe hbe-input-label-content hbe-input-label-content-default">Hey, password is required here.</span>      </label>    </div>  </div></div><script data-pjax src="/lib/hbe.js"></script><link href="/css/hbe.style.css" rel="stylesheet" type="text/css">]]></content>
      
      
      <categories>
          
          <category> LaTeX </category>
          
      </categories>
      
      
        <tags>
            
            <tag> LaTeX </tag>
            
        </tags>
      
    </entry>
    
    
    
    <entry>
      <title>Hello World</title>
      <link href="/2024/07/29/hello-world/"/>
      <url>/2024/07/29/hello-world/</url>
      
        <content type="html"><![CDATA[<p>Welcome to <a href="https://hexo.io/">Hexo</a>! This is your very first post. Check <a href="https://hexo.io/docs/">documentation</a> for more info. If you get any problems when using Hexo, you can find the answer in <a href="https://hexo.io/docs/troubleshooting.html">troubleshooting</a> or you can ask me on <a href="https://github.com/hexojs/hexo/issues">GitHub</a>.</p><h2 id="quick-start"><a class="markdownIt-Anchor" href="#quick-start"></a> Quick Start</h2><h3 id="create-a-new-post"><a class="markdownIt-Anchor" href="#create-a-new-post"></a> Create a new post</h3><pre class="highlight"><code class="bash">$ hexo new <span class="hljs-string">&quot;My New Post&quot;</span></code></pre><p>More info: <a href="https://hexo.io/docs/writing.html">Writing</a></p><h3 id="run-server"><a class="markdownIt-Anchor" href="#run-server"></a> Run server</h3><pre class="highlight"><code class="bash">$ hexo server</code></pre><p>More info: <a href="https://hexo.io/docs/server.html">Server</a></p><h3 id="generate-static-files"><a class="markdownIt-Anchor" href="#generate-static-files"></a> Generate static files</h3><pre class="highlight"><code class="bash">$ hexo generate</code></pre><p>More info: <a href="https://hexo.io/docs/generating.html">Generating</a></p><h3 id="deploy-to-remote-sites"><a class="markdownIt-Anchor" href="#deploy-to-remote-sites"></a> Deploy to remote sites</h3><pre class="highlight"><code class="bash">$ hexo deploy</code></pre><p>More info: <a href="https://hexo.io/docs/one-command-deployment.html">Deployment</a></p>]]></content>
      
      
      
    </entry>
    
    
    
    <entry>
      <title>[Theme] Readme</title>
      <link href="/2023/01/10/Cutie%20Document/"/>
      <url>/2023/01/10/Cutie%20Document/</url>
      
        <content type="html"><![CDATA[<h1 id="theme-cutie"><a class="markdownIt-Anchor" href="#theme-cutie"></a> Theme Cutie</h1><p><img src="https://cdn.jsdelivr.net/gh/yz-hs/PicGo/intro4.png" alt="" /></p><p><strong>This theme is still under development and maintenance. If there are bugs, optimization requirements, etc., you can Issue, PR or leave a message in <a href="/pages/chat">here</a>.</strong></p><p>中文版文档，本主题的完整说明：<a href="/2020/08/22/Cutie%20%E4%BD%BF%E7%94%A8%E6%96%87%E6%A1%A3/">Chinese version</a>。</p><span id="more"></span><h2 id="background"><a class="markdownIt-Anchor" href="#background"></a> Background</h2><p>In order to learn relevant knowledge and make a good-looking theme, I made this. The layout of the theme is like <a href="https://www.ihewro.com/archives/489/">handsome</a>, a typecho theme. Most of the codes were originally programmed by me, while the rest were from the Internet, for instance, <a href="https://mkblog.cn/">mkBlog</a>.</p><h2 id="install"><a class="markdownIt-Anchor" href="#install"></a> Install</h2><p>You can install it by running this command:</p><pre class="highlight"><code class="bash">git <span class="hljs-built_in">clone</span> https://github.com/yz-hs/cutie.git</code></pre><h2 id="usage"><a class="markdownIt-Anchor" href="#usage"></a> Usage</h2><p>Firstly, you need to set the value of <code>theme</code> in the blog configuration file (<code>/path-to-your-blog/_config.yml</code>) to <code>cutie</code> to enable this theme.</p><p>Then, you need to read documents to further configure the theme and learn more about this theme. Most of them are in Chinese.</p><p><strong>This theme relies on some plug-ins. Please be sure to read the <a href="/2020/08/22/Cutie%20%E4%BD%BF%E7%94%A8%E6%96%87%E6%A1%A3/">Chinese version</a> first.</strong></p><h2 id="demo"><a class="markdownIt-Anchor" href="#demo"></a> Demo</h2><p><a href="https://Yurchiu.github.io/">https://Yurchiu.github.io/</a></p><h2 id="maintainers"><a class="markdownIt-Anchor" href="#maintainers"></a> Maintainers</h2><p>@<a href="https://github.com/Yurchiu">Yurchiu</a></p><h2 id="contributors"><a class="markdownIt-Anchor" href="#contributors"></a> Contributors</h2><p>@<a href="https://github.com/Yurchiu">Yurchiu</a></p><p>@<a href="https://github.com/tsxc-github">tsxc</a></p><p>Feel free to dive in! Open an issue or submit PRs.</p><h2 id="license"><a class="markdownIt-Anchor" href="#license"></a> License</h2><p>MIT</p>]]></content>
      
      
      <categories>
          
          <category> Cutie </category>
          
      </categories>
      
      
        <tags>
            
            <tag> Hexo </tag>
            
            <tag> Cutie </tag>
            
        </tags>
      
    </entry>
    
    
    
    <entry>
      <title>Cutie 使用文档</title>
      <link href="/2020/08/22/Cutie%20%E4%BD%BF%E7%94%A8%E6%96%87%E6%A1%A3/"/>
      <url>/2020/08/22/Cutie%20%E4%BD%BF%E7%94%A8%E6%96%87%E6%A1%A3/</url>
      
        <content type="html"><![CDATA[<h1 id="cutie"><a class="markdownIt-Anchor" href="#cutie"></a> Cutie</h1><h2 id="介绍"><a class="markdownIt-Anchor" href="#介绍"></a> 介绍</h2><p><img src="https://cdn.jsdelivr.net/gh/yz-hs/PicGo/intro4.png" alt="" /></p><p>Cutie 是一个 Hexo 主题。本主题由 <a href="https://github.com/Yurchiu/">Yurchiu</a> 制作，外观借鉴了 <a href="https://www.ihewro.com/archives/489/">handsome</a> 主题，部分代码借鉴自 <a href="https://mkblog.cn/">mkBlog</a>。本主题可在 <a href="https://github.com/yz-hs/Cutie">Github</a> 获取。</p><p>本主题依赖一些插件。请务必先看本文档，否则无法部署成功。</p><p><strong>本主题仍在开发维护中。若有 Bug、优化需求等，可 Issue、PR 或在 <a href="https://Yurchiu.github.io/somepage/chat/">https://Yurchiu.github.io/somepage/chat/</a> 留言</strong>。</p><span id="more"></span><h2 id="特点"><a class="markdownIt-Anchor" href="#特点"></a> 特点</h2><ul><li>自适应。</li><li>支持多个第三方评论平台（Gitalk，Livere，Giscus 等等）。</li><li>几种配色（<code>std</code>、<code>white</code>、<code>night</code>，本 Blog 使用 <code>white</code> 配色）与三种特殊样式。</li><li>中英文支持。</li><li>Pjax 支持。</li><li>其他的许多特性！</li></ul><h2 id="todo-list"><a class="markdownIt-Anchor" href="#todo-list"></a> Todo List</h2><div class='checkbox'><input type="checkbox" />            <p>在未来优化整体框架。</p>            </div><h2 id="首次使用"><a class="markdownIt-Anchor" href="#首次使用"></a> 首次使用</h2><h3 id="dependencies"><a class="markdownIt-Anchor" href="#dependencies"></a> Dependencies</h3><ol><li><p>“RSS”依赖 <code>hexo-generator-feed</code> 插件来运行。 请运行 <code>npm install hexo-generator-feed</code>。</p></li><li><p>“wordcount”依赖 <code>hexo-wordcount</code> 插件来运行。 请运行 <code>npm i hexo-wordcount --save</code>。</p></li><li><p>“blog-encrypt”依赖 <code>hexo-blog-encrypt</code> 插件来运行。 请运行 <code>npm install --save hexo-blog-encrypt</code>。其使用方法，参见 <a href="https://github.com/D0n9X1n/hexo-blog-encrypt/blob/master/ReadMe.zh.md">https://github.com/D0n9X1n/hexo-blog-encrypt/blob/master/ReadMe.zh.md</a>。</p></li><li><p>“Search”依赖 <code>hexo-generator-search</code>。请运行 <code>npm install hexo-generator-search --save</code>。其使用方法，参见 <a href="https://github.com/wzpan/hexo-generator-search">https://github.com/wzpan/hexo-generator-search</a>。</p><p>设置站内搜索页的方法是：在 front-matter 加上 <code>layout: search</code>。</p></li><li><p>数学公式的依赖见下文。</p></li><li><p>推荐安装插件： <code>hexo-neat</code> <code>hexo-generator-sitemap</code> <code>hexo-butterfly-tag-plugins-plus</code>。</p></li></ol><h3 id="代码高亮"><a class="markdownIt-Anchor" href="#代码高亮"></a> 代码高亮</h3><p>去掉博客配置文件的自带代码高亮即可。</p><h3 id="数学公式"><a class="markdownIt-Anchor" href="#数学公式"></a> 数学公式</h3><p>本主题不内置对数学公式的支持（但已经包含 CSS）。请按照以下方法来支持数学公式：</p><p>Git bash 运行：</p><pre class="highlight"><code class="bash">npm un hexo-renderer-markednpm i hexo-renderer-markdown-it-plusnpm i katexnpm i @andatoshiki/markdown-it-katex</code></pre><p>在博客配置文件写入：</p><pre class="highlight"><code class="yaml"><span class="hljs-attr">markdown_it_plus:</span>  <span class="hljs-attr">plugins:</span>    <span class="hljs-bullet">-</span> <span class="hljs-attr">plugin:</span>      <span class="hljs-attr">name:</span> <span class="hljs-string">&#x27;@iktakahiro/markdown-it-katex&#x27;</span>      <span class="hljs-attr">enable:</span> <span class="hljs-literal">false</span>    <span class="hljs-bullet">-</span> <span class="hljs-attr">plugin:</span>      <span class="hljs-attr">name:</span> <span class="hljs-string">&#x27;@andatoshiki/markdown-it-katex&#x27;</span>      <span class="hljs-attr">enable:</span> <span class="hljs-literal">true</span></code></pre><p>记得 <code>hexo clean</code>。</p><h3 id="添加文章置顶"><a class="markdownIt-Anchor" href="#添加文章置顶"></a> 添加文章置顶</h3><h4 id="方法一"><a class="markdownIt-Anchor" href="#方法一"></a> 方法一</h4><p>请在 <code>path-to-your-blog/node_modules/hexo-generator-index/lib/generator.js</code> 的合适位置添加如下代码：</p><pre class="highlight"><code class="js">posts.<span class="hljs-property">data</span> = posts.<span class="hljs-property">data</span>.<span class="hljs-title function_">sort</span>(<span class="hljs-keyword">function</span>(<span class="hljs-params">a, b</span>) &#123;      <span class="hljs-keyword">if</span>(a.<span class="hljs-property">top</span> &amp;&amp; b.<span class="hljs-property">top</span>) &#123;          <span class="hljs-keyword">if</span>(a.<span class="hljs-property">top</span> == b.<span class="hljs-property">top</span>) <span class="hljs-keyword">return</span> b.<span class="hljs-property">date</span> - a.<span class="hljs-property">date</span>;          <span class="hljs-keyword">else</span> <span class="hljs-keyword">return</span> b.<span class="hljs-property">top</span> - a.<span class="hljs-property">top</span>;      &#125;      <span class="hljs-keyword">else</span> <span class="hljs-keyword">if</span>(a.<span class="hljs-property">top</span> &amp;&amp; !b.<span class="hljs-property">top</span>) &#123;          <span class="hljs-keyword">return</span> -<span class="hljs-number">1</span>;      &#125;      <span class="hljs-keyword">else</span> <span class="hljs-keyword">if</span>(!a.<span class="hljs-property">top</span> &amp;&amp; b.<span class="hljs-property">top</span>) &#123;          <span class="hljs-keyword">return</span> <span class="hljs-number">1</span>;      &#125;      <span class="hljs-keyword">else</span> <span class="hljs-keyword">return</span> b.<span class="hljs-property">date</span> - a.<span class="hljs-property">date</span>;  &#125;);</code></pre><p>如代码所示，“文章置顶” 原理是按照文章的 top 值排序，并加上 “置顶” 标签。</p><h4 id="方法二"><a class="markdownIt-Anchor" href="#方法二"></a> 方法二</h4><p>卸载系统自带的排序插件：</p><pre class="highlight"><code class="swift">npm uninstall hexo<span class="hljs-operator">-</span>generator<span class="hljs-operator">-</span>index</code></pre><p>添加替代插件：</p><pre class="highlight"><code class="swift">npm install hexo<span class="hljs-operator">-</span>generator<span class="hljs-operator">-</span>index<span class="hljs-operator">-</span>pin<span class="hljs-operator">-</span>top <span class="hljs-operator">--</span>save</code></pre><p>同样按照文章 top 值排序。</p><h2 id="页面配置"><a class="markdownIt-Anchor" href="#页面配置"></a> 页面配置</h2><h3 id="tagscategories"><a class="markdownIt-Anchor" href="#tagscategories"></a> Tags/Categories</h3><p>设置“标签页面”或“分类页面”的方法是：在 front-matter 加上 <code>layout: tags/categories</code>。</p><h3 id="search"><a class="markdownIt-Anchor" href="#search"></a> Search</h3><p>本主题支持两种搜索方式：站内搜索和站外搜索。</p><p>添加“搜索页面”方法：新建一个 page，在 front-matter 加上 <code>layout: search</code>。依赖 <code>hexo-generator-search</code>。</p><p>这里的站外搜索使用必应搜索。对于其他搜索引擎，可以自己更改。</p><p><strong>请注意：填入 search.xml 的链接时，一定要加上博客配置文件中的 url</strong>。</p><h3 id="setting"><a class="markdownIt-Anchor" href="#setting"></a> Setting</h3><p>本主题支持“设置页面”，供访者自主设置主题样式、特殊效果等。</p><p>添加“设置页面”方法：新建一个 page，在 front-matter 加上 <code>layout: settings</code>。</p><h3 id="mood"><a class="markdownIt-Anchor" href="#mood"></a> Mood</h3><p>本主题支持“说说页面”。</p><p>添加“说说页面”方法：新建一个 page，添加如下代码：</p><pre class="highlight"><code class="html">title: 说说---<span class="hljs-tag">&lt;<span class="hljs-name">script</span>&gt;</span><span class="language-javascript">  $(<span class="hljs-keyword">function</span>(<span class="hljs-params"></span>)&#123;$(<span class="hljs-string">&quot;info&quot;</span>).<span class="hljs-title function_">prepend</span>(<span class="hljs-string">&quot;&lt;i class=&#x27;fa fa-clock-o&#x27;&gt;&lt;/i&gt; &quot;</span>);&#125;);</span><span class="hljs-tag">&lt;/<span class="hljs-name">script</span>&gt;</span><span class="hljs-tag">&lt;<span class="hljs-name">ul</span> <span class="hljs-attr">class</span>=<span class="hljs-string">&quot;mood-list&quot;</span>&gt;</span><span class="hljs-tag">&lt;<span class="hljs-name">li</span>&gt;</span><span class="hljs-comment">&lt;!-- 头像，使用 img 标签 --&gt;</span>&quot;<span class="hljs-tag">&lt;<span class="hljs-name">text</span>&gt;</span><span class="hljs-tag">&lt;<span class="hljs-name">inner</span>&gt;</span><span class="hljs-comment">&lt;!-- 说说内容 --&gt;</span><span class="hljs-tag">&lt;<span class="hljs-name">info</span>&gt;</span><span class="hljs-comment">&lt;!-- 说说发表时间 --&gt;</span><span class="hljs-tag">&lt;/<span class="hljs-name">info</span>&gt;</span><span class="hljs-tag">&lt;/<span class="hljs-name">inner</span>&gt;</span><span class="hljs-tag">&lt;/<span class="hljs-name">text</span>&gt;</span><span class="hljs-tag">&lt;/<span class="hljs-name">li</span>&gt;</span>...<span class="hljs-comment">&lt;!-- 不断重复第 10-12 行 --&gt;</span><span class="hljs-tag">&lt;/<span class="hljs-name">ul</span>&gt;</span></code></pre><h3 id="404-page"><a class="markdownIt-Anchor" href="#404-page"></a> 404 page</h3><p>在博客配置文件写入：</p><pre class="highlight"><code class="yaml"><span class="hljs-attr">skip_render:</span>  <span class="hljs-bullet">-</span> <span class="hljs-string">&#x27;404.html&#x27;</span></code></pre><p>之后，把主题文件夹下的 <code>_doc/404.html</code> 置入博客 <code>source</code> 文件夹下（不是主题的 <code>source</code> 文件夹）。</p><p>或者，创建新 page 作为 404 页面。在 front-matter 写入 <code>permalink: /404.html</code>。</p><h2 id="其他"><a class="markdownIt-Anchor" href="#其他"></a> 其他</h2><h3 id="front-matter"><a class="markdownIt-Anchor" href="#front-matter"></a> Front-matter</h3><p>本主题可设置如下 front-matter：</p><ul><li><code>discomments: true</code>——是否禁止评论。</li><li><code>top: [number]</code>——设置文章排列优先级。注意，参看“添加文章置顶”部分来获取启用方式。</li><li><code>author: [list/array of string]</code>——设置作者（支持多作者）。默认为博客根目录中 <code>config_.yml</code> 中的内容。</li></ul><p>下面是关于文章卡片的。文章卡片指在首页显示的含文章摘要的文本框。设置了背景图片，称为图片文章卡片；否则称为文字文章卡片。</p><p>图片文章卡片：</p><ul><li><code>bgimg: /path/to/your/pic.img</code>——设置文章卡片背景图片。</li><li><code>height: [string]</code>——设定文章卡片高度（数字+单位）。默认 <code>30em</code>，亦可在 <code>_config.yml</code> 修改默认值。</li></ul><p>文字文章卡片：</p><ul><li><code>faname: [string]</code>——设置文章卡片的 Font Awesome 图标。默认是 <code>fa-file-text-o</code>。</li><li><code>disfa: true</code>——禁用文章卡片的 Font Awesome 图标。</li></ul><p>注意：<code>bgimg</code> 和 <code>faname</code> 冲突。<code>bgimg</code> 优先。</p><h3 id="禁用-devtools"><a class="markdownIt-Anchor" href="#禁用-devtools"></a> 禁用 Devtools</h3><p>在需禁用 Devtools 的页面加入如下代码：</p><pre class="highlight"><code class="html"><span class="hljs-tag">&lt;<span class="hljs-name">script</span> <span class="hljs-attr">src</span>=<span class="hljs-string">&quot;/js/devtools.js&quot;</span>&gt;</span><span class="hljs-tag">&lt;/<span class="hljs-name">script</span>&gt;</span></code></pre><p>功能：</p><ul><li>当使用 Devtools 时，跳转网页；</li><li>禁止右键，禁止选中文本。</li></ul><h3 id="custom"><a class="markdownIt-Anchor" href="#custom"></a> Custom</h3><p>为了避免自主修改主题源码造成的麻烦，自定义 css 及 js 时，可以在 <code>/source/custom</code> 文件夹下自主添加所需要的 css 和 js 代码。</p><p>主题只会引用其中的 <code>custom.css</code> 和 <code>custom.js</code>。</p><h3 id="短代码效果"><a class="markdownIt-Anchor" href="#短代码效果"></a> 短代码效果</h3><p>理论上 Bootstrap5 中的所有都可以使用。另外，本主题还包括下面的一些东西。</p><details><summary>混乱文字</summary><p><strong>代码</strong></p><pre class="highlight"><code class="html"><span class="hljs-tag">&lt;<span class="hljs-name">div</span> <span class="hljs-attr">class</span>=<span class="hljs-string">&quot;chaffle&quot;</span> <span class="hljs-attr">data-lang</span>=<span class="hljs-string">&quot;zh&quot;</span>&gt;</span>“她好可爱啊！”<span class="hljs-tag">&lt;/<span class="hljs-name">div</span>&gt;</span></code></pre><p><strong>效果</strong></p><div class="chaffle" data-lang="zh">“她好可爱啊！”</div></details><hr /><details><summary>展开折叠</summary><p><strong>代码</strong></p><pre class="highlight"><code class="html"><span class="hljs-tag">&lt;<span class="hljs-name">details</span>&gt;</span>    <span class="hljs-tag">&lt;<span class="hljs-name">summary</span>&gt;</span>请点击<span class="hljs-tag">&lt;/<span class="hljs-name">summary</span>&gt;</span>    <span class="hljs-tag">&lt;<span class="hljs-name">p</span>&gt;</span>就算风吹散了冰雪，想念也会留存下来。<span class="hljs-tag">&lt;<span class="hljs-name">br</span> /&gt;</span>鼠标点击会展开此内容。<span class="hljs-tag">&lt;/<span class="hljs-name">p</span>&gt;</span><span class="hljs-tag">&lt;/<span class="hljs-name">details</span>&gt;</span>防止混淆的文字。</code></pre><p><strong>效果</strong></p><details>    <summary>请点击</summary>    <p>就算风吹散了冰雪，想念也会留存下来。<br />鼠标点击会展开此内容。</p></details><p>防止混淆的文字。</p></details><hr /><details><summary>黑幕类</summary><p><strong>代码</strong></p><pre class="highlight"><code class="html">提示：<span class="hljs-tag">&lt;<span class="hljs-name">shady</span> <span class="hljs-attr">title</span>=<span class="hljs-string">&quot;你知道的太多了&quot;</span>&gt;</span>就算风吹散了冰雪，想念也会留存下来。鼠标移在此上会显示此内容<span class="hljs-tag">&lt;/<span class="hljs-name">shady</span>&gt;</span>。仿自：[萌娘百科](https://zh.moegirl.org.cn/)。由于本句话被高度加密，即使使用小刀或者<span class="hljs-tag">&lt;<span class="hljs-name">black</span> <span class="hljs-attr">title</span>=<span class="hljs-string">&quot;你不知道的太多了&quot;</span>&gt;</span>除了被选中<span class="hljs-tag">&lt;/<span class="hljs-name">black</span>&gt;</span>也无法划开屏幕上的部分黑幕。提示：<span class="hljs-tag">&lt;<span class="hljs-name">blur</span>&gt;</span>没错，你近视了。<span class="hljs-tag">&lt;/<span class="hljs-name">blur</span>&gt;</span><span class="hljs-tag">&lt;<span class="hljs-name">invsb</span>&gt;</span>以至于你都看不见这句话。<span class="hljs-tag">&lt;/<span class="hljs-name">invsb</span>&gt;</span><span class="hljs-tag">&lt;<span class="hljs-name">blur</span>&gt;</span><span class="hljs-tag">&lt;<span class="hljs-name">fieldset</span>&gt;</span>    <span class="hljs-tag">&lt;<span class="hljs-name">legend</span>&gt;</span>温馨提示<span class="hljs-tag">&lt;/<span class="hljs-name">legend</span>&gt;</span>    你需要换眼镜。请立即赶往附近的眼镜店。<span class="hljs-tag">&lt;/<span class="hljs-name">fieldset</span>&gt;</span><span class="hljs-tag">&lt;/<span class="hljs-name">blur</span>&gt;</span><span class="hljs-tag">&lt;<span class="hljs-name">invsb</span>&gt;</span><span class="hljs-tag">&lt;<span class="hljs-name">div</span> <span class="hljs-attr">class</span>=<span class="hljs-string">&quot;text-alert danger&quot;</span>&gt;</span><span class="hljs-tag">&lt;<span class="hljs-name">i</span> <span class="hljs-attr">class</span>=<span class="hljs-string">&quot;fa fa-info-circle&quot;</span>&gt;</span><span class="hljs-tag">&lt;/<span class="hljs-name">i</span>&gt;</span>不用换眼镜了。你已经看不见了。<span class="hljs-tag">&lt;/<span class="hljs-name">div</span>&gt;</span><span class="hljs-tag">&lt;/<span class="hljs-name">invsb</span>&gt;</span></code></pre><p><strong>效果</strong></p><p>提示：<shady title="你知道的太多了">就算风吹散了冰雪，想念也会留存下来。鼠标移在此上会显示此内容</shady>。仿自：<a href="https://zh.moegirl.org.cn/">萌娘百科</a>。</p><p>由于本句话被高度加密，即使使用小刀或者<black title="你不知道的太多了">除了被选中</black>也无法划开屏幕上的部分黑幕。</p><p>提示：<blur>没错，你近视了。</blur><invsb>以至于你都看不见这句话。</invsb></p><blur><fieldset>    <legend>温馨提示</legend>    你需要换眼镜。请立即赶往附近的眼镜店。</fieldset></blur><br><div class="alert alert-danger"><invsb><i class="fa fa-info-circle"></i>不用换眼镜了。你已经看不见了。</invsb></div></details><hr /><details><summary>弹窗</summary><p><strong>代码</strong></p><p>请访问文章：<a href="/b12321c1b85e/">SweetAlert 的使用</a>。</p><p><strong>效果</strong></p><p><button type="button" class="btn btn-outline-secondary" onclick="demo()" style="display:block;margin:0 auto">点击查看Demo</button></p><script>  function demo() {swal.fire({      title: "这是示例 >_<",      text: "⌈ Yurchiu 是一只可爱的 OIer。 ⌋",      icon: "success"}).then((result) => {swal.fire({  title: '请确认',  text: "如果你吊打了 Yurchiu，你会 AK IOI！ >_<",  showCancelButton: true,  confirmButtonText: '吊打她！',  cancelButtonText: '她这么可爱，算了！',  reverseButtons: true}).then((result) => {  if (result.value) {    swal.fire({      text: '你成功地吊打了 Yurchiu，并 AK 了 IOI，orz。',    })  } else if (    result.dismiss === Swal.DismissReason.cancel  ) {    swal.fire({      text: '你不屑于吊打 Yurchiu，而是 AK IOI，orz。',    })  }})})}</script></details><hr /><details><summary>提示框</summary><p><strong>代码</strong></p><pre class="highlight"><code class="html"><span class="hljs-tag">&lt;<span class="hljs-name">fieldset</span>&gt;</span>    <span class="hljs-tag">&lt;<span class="hljs-name">legend</span>&gt;</span>温馨提示<span class="hljs-tag">&lt;/<span class="hljs-name">legend</span>&gt;</span>    <span class="hljs-tag">&lt;<span class="hljs-name">p</span>&gt;</span>提示点什么好呢？让我想想……<span class="hljs-tag">&lt;/<span class="hljs-name">p</span>&gt;</span><span class="hljs-tag">&lt;/<span class="hljs-name">fieldset</span>&gt;</span></code></pre><p><strong>效果</strong></p><fieldset>    <legend>温馨提示</legend>    <p>提示点什么好呢？让我想想……</p></fieldset></details><hr /><details><summary>分栏</summary><p><strong>代码</strong></p><p>html（建议不要复制这个）:</p><pre class="highlight"><code class="html"><span class="hljs-tag">&lt;<span class="hljs-name">div</span> <span class="hljs-attr">class</span>=<span class="hljs-string">&quot;tab-page&quot;</span>&gt;</span> <span class="hljs-tag">&lt;<span class="hljs-name">div</span> <span class="hljs-attr">class</span>=<span class="hljs-string">&quot;tabTitle&quot;</span>&gt;</span>  <span class="hljs-tag">&lt;<span class="hljs-name">ul</span>&gt;</span>    <span class="hljs-tag">&lt;<span class="hljs-name">li</span> <span class="hljs-attr">class</span>=<span class="hljs-string">&quot;current&quot;</span>&gt;</span>默认<span class="hljs-tag">&lt;/<span class="hljs-name">li</span>&gt;</span>    <span class="hljs-tag">&lt;<span class="hljs-name">li</span>&gt;</span>文字<span class="hljs-tag">&lt;/<span class="hljs-name">li</span>&gt;</span>    <span class="hljs-tag">&lt;<span class="hljs-name">li</span>&gt;</span>其他<span class="hljs-tag">&lt;/<span class="hljs-name">li</span>&gt;</span>  <span class="hljs-tag">&lt;/<span class="hljs-name">ul</span>&gt;</span> <span class="hljs-tag">&lt;/<span class="hljs-name">div</span>&gt;</span> <span class="hljs-tag">&lt;<span class="hljs-name">div</span> <span class="hljs-attr">class</span>=<span class="hljs-string">&quot;tabContent&quot;</span>&gt;</span>  <span class="hljs-tag">&lt;<span class="hljs-name">div</span>&gt;</span>请在 `tabTitle` 的第一个的 `li` 标签加上 `current` 类，除了 `tabContent` 的第一个 `div` 标签都加 `hide` 类。你可以加不止三个标签。不过太长会溢出。以下是代码纯净版（建议复制这个）：​```html<span class="hljs-tag">&lt;<span class="hljs-name">div</span> <span class="hljs-attr">class</span>=<span class="hljs-string">&quot;tab-page&quot;</span>&gt;</span> <span class="hljs-tag">&lt;<span class="hljs-name">div</span> <span class="hljs-attr">class</span>=<span class="hljs-string">&quot;tabTitle&quot;</span>&gt;</span>  <span class="hljs-tag">&lt;<span class="hljs-name">ul</span>&gt;</span>    <span class="hljs-tag">&lt;<span class="hljs-name">li</span> <span class="hljs-attr">class</span>=<span class="hljs-string">&quot;current&quot;</span>&gt;</span><span class="hljs-tag">&lt;/<span class="hljs-name">li</span>&gt;</span>    <span class="hljs-tag">&lt;<span class="hljs-name">li</span>&gt;</span><span class="hljs-tag">&lt;/<span class="hljs-name">li</span>&gt;</span>    <span class="hljs-tag">&lt;<span class="hljs-name">li</span>&gt;</span><span class="hljs-tag">&lt;/<span class="hljs-name">li</span>&gt;</span>  <span class="hljs-tag">&lt;/<span class="hljs-name">ul</span>&gt;</span> <span class="hljs-tag">&lt;/<span class="hljs-name">div</span>&gt;</span> <span class="hljs-tag">&lt;<span class="hljs-name">div</span> <span class="hljs-attr">class</span>=<span class="hljs-string">&quot;tabContent&quot;</span>&gt;</span>  <span class="hljs-tag">&lt;<span class="hljs-name">div</span>&gt;</span><span class="hljs-tag">&lt;/<span class="hljs-name">div</span>&gt;</span>  <span class="hljs-tag">&lt;<span class="hljs-name">div</span> <span class="hljs-attr">class</span>=<span class="hljs-string">&quot;hide&quot;</span>&gt;</span><span class="hljs-tag">&lt;/<span class="hljs-name">div</span>&gt;</span>  <span class="hljs-tag">&lt;<span class="hljs-name">div</span> <span class="hljs-attr">class</span>=<span class="hljs-string">&quot;hide&quot;</span>&gt;</span><span class="hljs-tag">&lt;/<span class="hljs-name">div</span>&gt;</span> <span class="hljs-tag">&lt;/<span class="hljs-name">div</span>&gt;</span><span class="hljs-tag">&lt;/<span class="hljs-name">div</span>&gt;</span>​```<span class="hljs-tag">&lt;/<span class="hljs-name">div</span>&gt;</span><span class="hljs-tag">&lt;<span class="hljs-name">div</span> <span class="hljs-attr">class</span>=<span class="hljs-string">&quot;hide&quot;</span>&gt;</span>You AK IOI!You AK IOI!You AK IOI!You AK IOI!You AK IOI!You AK IOI!You AK IOI!You AK IOI!You AK IOI!You AK IOI!You AK IOI!You AK IOI!You AK IOI!You AK IOI!You AK IOI!You AK IOI!You AK IOI!You AK IOI!You AK IOI!You AK IOI!You AK IOI!You AK IOI!You AK IOI!You AK IOI!You AK IOI!You AK IOI!You AK IOI!You AK IOI!You AK IOI!You AK IOI!You AK IOI!You AK IOI!You AK IOI!You AK IOI!You AK IOI!You AK IOI!You AK IOI!You AK IOI!You AK IOI!You AK IOI!You AK IOI!You AK IOI!You AK IOI!You AK IOI!You AK IOI!You AK IOI!You AK IOI!You AK IOI!You AK IOI!You AK IOI!You AK IOI!You AK IOI!You AK IOI!You AK IOI!You AK IOI!You AK IOI!You AK IOI!You AK IOI!<span class="hljs-tag">&lt;/<span class="hljs-name">div</span>&gt;</span><span class="hljs-tag">&lt;<span class="hljs-name">div</span> <span class="hljs-attr">class</span>=<span class="hljs-string">&quot;hide&quot;</span>&gt;</span><span class="hljs-tag">&lt;<span class="hljs-name">details</span>&gt;</span>    <span class="hljs-tag">&lt;<span class="hljs-name">summary</span>&gt;</span>请点击<span class="hljs-tag">&lt;/<span class="hljs-name">summary</span>&gt;</span>就算风吹散了冰雪，想念也会留存下来。<span class="hljs-tag">&lt;/<span class="hljs-name">details</span>&gt;</span>另一温馨提示：<span class="hljs-tag">&lt;<span class="hljs-name">shady</span> <span class="hljs-attr">title</span>=<span class="hljs-string">&quot;你知道的太多了&quot;</span>&gt;</span>是的，在标签页中也可加入HTML元素！<span class="hljs-tag">&lt;/<span class="hljs-name">shady</span>&gt;</span><span class="hljs-tag">&lt;/<span class="hljs-name">div</span>&gt;</span><span class="hljs-tag">&lt;/<span class="hljs-name">div</span>&gt;</span><span class="hljs-tag">&lt;/<span class="hljs-name">div</span>&gt;</span></code></pre><p>javascript:</p><pre class="highlight"><code class="javascript">$(<span class="hljs-keyword">function</span>(<span class="hljs-params"></span>)&#123;      <span class="hljs-keyword">var</span> ali = $(<span class="hljs-string">&#x27;tabTitle&gt;ul&gt;li&#x27;</span>);      <span class="hljs-keyword">var</span> aDiv = $(<span class="hljs-string">&#x27;tabContent&gt;div&#x27;</span>);      <span class="hljs-keyword">var</span> timeId = <span class="hljs-literal">null</span>;      ali.<span class="hljs-title function_">click</span>(<span class="hljs-keyword">function</span>(<span class="hljs-params"></span>)&#123;        <span class="hljs-keyword">var</span> _this = $(<span class="hljs-variable language_">this</span>);        timeId = <span class="hljs-built_in">setTimeout</span>(<span class="hljs-keyword">function</span>(<span class="hljs-params"></span>)&#123;          _this.<span class="hljs-title function_">addClass</span>(<span class="hljs-string">&#x27;current&#x27;</span>).<span class="hljs-title function_">siblings</span>().<span class="hljs-title function_">removeClass</span>(<span class="hljs-string">&#x27;current&#x27;</span>);          <span class="hljs-keyword">var</span> index = _this.<span class="hljs-title function_">index</span>();          aDiv.<span class="hljs-title function_">eq</span>(index).<span class="hljs-title function_">show</span>().<span class="hljs-title function_">siblings</span>().<span class="hljs-title function_">hide</span>();        &#125;);      &#125;);  &#125;);</code></pre><p><strong>效果</strong></p><div class="tab-page"> <div class="tabTitle">  <ul>    <li class="current">默认</li>    <li>文字</li>    <li>其他</li>  </ul> </div> <div class="tabContent">  <div>请在 `tabTitle` 的第一个的 `li` 标签加上 `current` 类，除了 `tabContent` 的第一个 `div` 标签都加 `hide` 类。<p>你可以加不止三个标签。不过太长会溢出。</p><p>以下是代码纯净版（建议复制这个）：</p><pre class="highlight"><code class="html"><span class="hljs-tag">&lt;<span class="hljs-name">div</span> <span class="hljs-attr">class</span>=<span class="hljs-string">&quot;tab-page&quot;</span>&gt;</span> <span class="hljs-tag">&lt;<span class="hljs-name">div</span> <span class="hljs-attr">class</span>=<span class="hljs-string">&quot;tabTitle&quot;</span>&gt;</span>  <span class="hljs-tag">&lt;<span class="hljs-name">ul</span>&gt;</span>    <span class="hljs-tag">&lt;<span class="hljs-name">li</span> <span class="hljs-attr">class</span>=<span class="hljs-string">&quot;current&quot;</span>&gt;</span><span class="hljs-tag">&lt;/<span class="hljs-name">li</span>&gt;</span>    <span class="hljs-tag">&lt;<span class="hljs-name">li</span>&gt;</span><span class="hljs-tag">&lt;/<span class="hljs-name">li</span>&gt;</span>    <span class="hljs-tag">&lt;<span class="hljs-name">li</span>&gt;</span><span class="hljs-tag">&lt;/<span class="hljs-name">li</span>&gt;</span>  <span class="hljs-tag">&lt;/<span class="hljs-name">ul</span>&gt;</span> <span class="hljs-tag">&lt;/<span class="hljs-name">div</span>&gt;</span> <span class="hljs-tag">&lt;<span class="hljs-name">div</span> <span class="hljs-attr">class</span>=<span class="hljs-string">&quot;tabContent&quot;</span>&gt;</span>  <span class="hljs-tag">&lt;<span class="hljs-name">div</span>&gt;</span><span class="hljs-tag">&lt;/<span class="hljs-name">div</span>&gt;</span>  <span class="hljs-tag">&lt;<span class="hljs-name">div</span> <span class="hljs-attr">class</span>=<span class="hljs-string">&quot;hide&quot;</span>&gt;</span><span class="hljs-tag">&lt;/<span class="hljs-name">div</span>&gt;</span>  <span class="hljs-tag">&lt;<span class="hljs-name">div</span> <span class="hljs-attr">class</span>=<span class="hljs-string">&quot;hide&quot;</span>&gt;</span><span class="hljs-tag">&lt;/<span class="hljs-name">div</span>&gt;</span> <span class="hljs-tag">&lt;/<span class="hljs-name">div</span>&gt;</span><span class="hljs-tag">&lt;/<span class="hljs-name">div</span>&gt;</span></code></pre></div><div class="hide"><p>You AK IOI!You AK IOI!You AK IOI!You AK IOI!You AK IOI!You AK IOI!You AK IOI!You AK IOI!You AK IOI!You AK IOI!You AK IOI!You AK IOI!You AK IOI!You AK IOI!You AK IOI!You AK IOI!You AK IOI!You AK IOI!You AK IOI!You AK IOI!You AK IOI!You AK IOI!You AK IOI!You AK IOI!You AK IOI!You AK IOI!You AK IOI!You AK IOI!You AK IOI!You AK IOI!You AK IOI!You AK IOI!You AK IOI!You AK IOI!You AK IOI!You AK IOI!You AK IOI!You AK IOI!You AK IOI!You AK IOI!You AK IOI!You AK IOI!You AK IOI!You AK IOI!You AK IOI!You AK IOI!You AK IOI!You AK IOI!You AK IOI!You AK IOI!You AK IOI!You AK IOI!You AK IOI!You AK IOI!You AK IOI!You AK IOI!You AK IOI!You AK IOI!</p></div><div class="hide"><details>    <summary>请点击</summary><p>就算风吹散了冰雪，想念也会留存下来。</p></details><p>另一温馨提示：<shady title="你知道的太多了">是的，在标签页中也可加入HTML元素！</shady></p></div></div></div><script>$(function(){      var ali = $('.tabTitle>ul>li');      var aDiv = $('.tabContent>div');      var timeId = null;      ali.click(function(){        var _this = $(this);        timeId = setTimeout(function(){          _this.addClass('current').siblings().removeClass('current');          var index = _this.index();          aDiv.eq(index).show().siblings().hide();        });      });  });</script></details><hr /><details><summary>标签</summary><p><strong>代码</strong></p><pre class="highlight"><code class="html">请在 `tabTitle` 的<span class="hljs-tag">&lt;<span class="hljs-name">tag</span>&gt;</span>第一个<span class="hljs-tag">&lt;/<span class="hljs-name">tag</span>&gt;</span>的 `li` 标签加上 `current` 类，除了 `tabContent` 的第一个 `div` 标签都加 `hide` 类。你可以加<span class="hljs-tag">&lt;<span class="hljs-name">tag</span>&gt;</span>不止三个<span class="hljs-tag">&lt;/<span class="hljs-name">tag</span>&gt;</span>标签。不过太长会<span class="hljs-tag">&lt;<span class="hljs-name">tag</span> <span class="hljs-attr">style</span>=<span class="hljs-string">&quot;background: orange&quot;</span>&gt;</span>溢出<span class="hljs-tag">&lt;/<span class="hljs-name">tag</span>&gt;</span>。`tag` 自动轮换四种背景色，当然也可以自己指定。</code></pre><p><strong>效果</strong></p><p>请在 <code>tabTitle</code> 的<tag>第一个</tag>的 <code>li</code> 标签加上 <code>current</code> 类，除了 <code>tabContent</code> 的第一个 <code>div</code> 标签都加 <code>hide</code> 类。</p><p>你可以加<tag>不止三个</tag>标签。不过太长会<tag style="background: orange">溢出</tag>。</p><p><code>tag</code> 自动轮换四种背景色，当然也可以自己指定。</p></details><hr /><details><summary>小号文字</summary>​    <p><strong>代码</strong></p><pre class="highlight"><code class="html">温馨提示：提示点什么好呢<span class="hljs-tag">&lt;<span class="hljs-name">small</span>&gt;</span>真的没有可提示的,,,<span class="hljs-tag">&lt;/<span class="hljs-name">small</span>&gt;</span></code></pre><p><strong>效果</strong></p><p>温馨提示：提示点什么好呢<small>真的没有可提示的,</small></p></details><hr /><details><summary>字幕</summary><p><strong>代码</strong></p><pre class="highlight"><code class="html"><span class="hljs-comment">&lt;!--滚动方向 direction 4个值 up down left right  默认从右向左--&gt;</span><span class="hljs-tag">&lt;<span class="hljs-name">marquee</span> <span class="hljs-attr">direction</span>=<span class="hljs-string">&quot;up&quot;</span>&gt;</span>我向上滚动<span class="hljs-tag">&lt;/<span class="hljs-name">marquee</span>&gt;</span><span class="hljs-comment">&lt;!--3个值 scroll-循环滚动 slide-只滚动一次 alternate-来回滚动  默认循环滚动--&gt;</span><span class="hljs-tag">&lt;<span class="hljs-name">marquee</span> <span class="hljs-attr">behavior</span>=<span class="hljs-string">&quot;slide&quot;</span>&gt;</span>我只滚动一次<span class="hljs-tag">&lt;/<span class="hljs-name">marquee</span>&gt;</span><span class="hljs-comment">&lt;!--值越大，滚动速度越快 一般5-10比较适宜消息观看--&gt;</span><span class="hljs-tag">&lt;<span class="hljs-name">marquee</span> <span class="hljs-attr">scrollamount</span>=<span class="hljs-string">&quot;20&quot;</span>&gt;</span>我是速度为20的滚动<span class="hljs-tag">&lt;/<span class="hljs-name">marquee</span>&gt;</span><span class="hljs-comment">&lt;!--值越大，滚动速度越慢，通常不设置--&gt;</span><span class="hljs-tag">&lt;<span class="hljs-name">marquee</span> <span class="hljs-attr">scrolldelay</span>=<span class="hljs-string">&quot;110&quot;</span>&gt;</span>我延迟滚动<span class="hljs-tag">&lt;/<span class="hljs-name">marquee</span>&gt;</span><span class="hljs-comment">&lt;!-- 默认值-1或infinite 表示无限循环滚动 loop=&quot;数值&quot; 表示滚动相应的次数--&gt;</span><span class="hljs-tag">&lt;<span class="hljs-name">marquee</span> <span class="hljs-attr">loop</span>=<span class="hljs-string">&quot;2&quot;</span>&gt;</span>我是loop循环滚动<span class="hljs-tag">&lt;/<span class="hljs-name">marquee</span>&gt;</span><span class="hljs-comment">&lt;!--宽100px 高90px 背景色为#f5f5f5的滚动区域--&gt;</span><span class="hljs-tag">&lt;<span class="hljs-name">marquee</span> <span class="hljs-attr">width</span>=<span class="hljs-string">&quot;100&quot;</span> <span class="hljs-attr">height</span>=<span class="hljs-string">&quot;90&quot;</span> <span class="hljs-attr">bgcolor</span>=<span class="hljs-string">&quot;#f5f5f5&quot;</span> &gt;</span>    <span class="hljs-tag">&lt;<span class="hljs-name">p</span>&gt;</span>我是滚动1<span class="hljs-tag">&lt;/<span class="hljs-name">p</span>&gt;</span>    <span class="hljs-tag">&lt;<span class="hljs-name">p</span>&gt;</span>我是滚动2<span class="hljs-tag">&lt;/<span class="hljs-name">p</span>&gt;</span>    <span class="hljs-tag">&lt;<span class="hljs-name">p</span>&gt;</span>我是滚动3<span class="hljs-tag">&lt;/<span class="hljs-name">p</span>&gt;</span><span class="hljs-tag">&lt;/<span class="hljs-name">marquee</span>&gt;</span><span class="hljs-comment">&lt;!--滚动空间 hspace-水平边距 vspace-垂直边距--&gt;</span><span class="hljs-tag">&lt;<span class="hljs-name">marquee</span> <span class="hljs-attr">width</span>=<span class="hljs-string">&quot;50&quot;</span> <span class="hljs-attr">hspace</span>=<span class="hljs-string">&quot;20&quot;</span> <span class="hljs-attr">vspace</span>=<span class="hljs-string">&quot;10&quot;</span> &gt;</span><span class="hljs-tag">&lt;<span class="hljs-name">p</span>&gt;</span>我是滚动1<span class="hljs-tag">&lt;/<span class="hljs-name">p</span>&gt;</span>    <span class="hljs-tag">&lt;<span class="hljs-name">p</span>&gt;</span>我是滚动2<span class="hljs-tag">&lt;/<span class="hljs-name">p</span>&gt;</span>    <span class="hljs-tag">&lt;<span class="hljs-name">p</span>&gt;</span>我是滚动3<span class="hljs-tag">&lt;/<span class="hljs-name">p</span>&gt;</span><span class="hljs-tag">&lt;/<span class="hljs-name">marquee</span>&gt;</span><span class="hljs-comment">&lt;!--鼠标悬停，滚动停止   鼠标离开，滚动继续--&gt;</span><span class="hljs-tag">&lt;<span class="hljs-name">marquee</span> <span class="hljs-attr">onmouseover</span>=<span class="hljs-string">&quot;this.stop()&quot;</span> <span class="hljs-attr">onmouseout</span>=<span class="hljs-string">&quot;this.start()&quot;</span>&gt;</span>    我是滚动<span class="hljs-tag">&lt;/<span class="hljs-name">marquee</span>&gt;</span></code></pre><p><strong>效果</strong></p><!--滚动方向 direction 4个值 up down left right  默认从右向左--><p><marquee direction="up">我向上滚动</marquee></p><!--3个值 scroll-循环滚动 slide-只滚动一次 alternate-来回滚动  默认循环滚动--><p><marquee behavior="slide">我只滚动一次</marquee></p><!--值越大，滚动速度越快 一般5-10比较适宜消息观看--><p><marquee scrollamount="20">我是速度为20的滚动</marquee></p><!--值越大，滚动速度越慢，通常不设置--><p><marquee scrolldelay="110">我延迟滚动</marquee></p><!-- 默认值-1或infinite 表示无限循环滚动 loop="数值" 表示滚动相应的次数--><p><marquee loop="2">我是loop循环滚动</marquee></p><!--宽100px 高90px 背景色为#f5f5f5的滚动区域--><marquee width="100" height="90" bgcolor="#f5f5f5" >    <p>我是滚动1</p>    <p>我是滚动2</p>    <p>我是滚动3</p></marquee><!--滚动空间 hspace-水平边距 vspace-垂直边距--><marquee width="50" hspace="20" vspace="10" ><p>我是滚动1</p>    <p>我是滚动2</p>    <p>我是滚动3</p></marquee><!--鼠标悬停，滚动停止   鼠标离开，滚动继续--><marquee onmouseover="this.stop()" onmouseout="this.start()"><p>我是滚动</p></marquee></details><hr /><details><summary>时间轴</summary><p><strong>代码</strong></p><pre class="highlight"><code class="html"><span class="hljs-tag">&lt;<span class="hljs-name">ol</span> <span class="hljs-attr">class</span>=<span class="hljs-string">&quot;timeline&quot;</span>&gt;</span>    <span class="hljs-tag">&lt;<span class="hljs-name">li</span>&gt;</span> <span class="hljs-tag">&lt;<span class="hljs-name">t</span>&gt;</span>时间轴<span class="hljs-tag">&lt;/<span class="hljs-name">t</span>&gt;</span> <span class="hljs-tag">&lt;/<span class="hljs-name">li</span>&gt;</span>    <span class="hljs-tag">&lt;<span class="hljs-name">li</span>&gt;</span> <span class="hljs-tag">&lt;<span class="hljs-name">b</span>&gt;</span>2020/13/32<span class="hljs-tag">&lt;/<span class="hljs-name">b</span>&gt;</span><span class="hljs-tag">&lt;<span class="hljs-name">p</span>&gt;</span>内容 1<span class="hljs-tag">&lt;/<span class="hljs-name">p</span>&gt;</span> <span class="hljs-tag">&lt;/<span class="hljs-name">li</span>&gt;</span>    <span class="hljs-tag">&lt;<span class="hljs-name">li</span>&gt;</span> <span class="hljs-tag">&lt;<span class="hljs-name">b</span>&gt;</span>2020/13/33<span class="hljs-tag">&lt;/<span class="hljs-name">b</span>&gt;</span><span class="hljs-tag">&lt;<span class="hljs-name">p</span>&gt;</span>内容 2<span class="hljs-tag">&lt;/<span class="hljs-name">p</span>&gt;</span> <span class="hljs-tag">&lt;/<span class="hljs-name">li</span>&gt;</span><span class="hljs-tag">&lt;/<span class="hljs-name">ol</span>&gt;</span></code></pre><p><strong>效果</strong></p><ol class="timeline">    <li> <t>时间轴</t> </li>    <li> <b>2020/13/32</b><p>内容 1</p> </li>    <li> <b>2020/13/33</b><p>内容 2</p> </li></ol></details><hr /><details><summary>注音</summary><p><strong>代码</strong></p><pre class="highlight"><code class="html"><span class="hljs-tag">&lt;<span class="hljs-name">ruby</span>&gt;</span>那 <span class="hljs-tag">&lt;<span class="hljs-name">rt</span>&gt;</span>n<span class="hljs-tag">&lt;/<span class="hljs-name">rt</span>&gt;</span>没 <span class="hljs-tag">&lt;<span class="hljs-name">rt</span>&gt;</span>m<span class="hljs-tag">&lt;/<span class="hljs-name">rt</span>&gt;</span>事 <span class="hljs-tag">&lt;<span class="hljs-name">rt</span>&gt;</span>s<span class="hljs-tag">&lt;/<span class="hljs-name">rt</span>&gt;</span>了 <span class="hljs-tag">&lt;<span class="hljs-name">rt</span>&gt;</span>l<span class="hljs-tag">&lt;/<span class="hljs-name">rt</span>&gt;</span><span class="hljs-tag">&lt;/<span class="hljs-name">ruby</span>&gt;</span><span class="hljs-tag">&lt;<span class="hljs-name">ruby</span>&gt;</span>那没事了 <span class="hljs-tag">&lt;<span class="hljs-name">rt</span>&gt;</span>nmsl<span class="hljs-tag">&lt;/<span class="hljs-name">rt</span>&gt;</span><span class="hljs-tag">&lt;/<span class="hljs-name">ruby</span>&gt;</span></code></pre><p><strong>效果</strong></p><ruby>那 <rt>n</rt>没 <rt>m</rt>事 <rt>s</rt>了 <rt>l</rt></ruby><ruby>那没事了 <rt>nmsl</rt></ruby></details><h3 id="外挂标签插件-third-party"><a class="markdownIt-Anchor" href="#外挂标签插件-third-party"></a> 外挂标签插件 <code>Third Party</code></h3><p>参见 <a href="https://akilar.top/posts/615e2dec/">https://akilar.top/posts/615e2dec/</a>。这里只展示效果。未列出的说明不支持，或不实用。通常会造成附近的 markdown 失效。</p><p>要想使用它，需要安装第三方插件 <code>hexo-butterfly-tag-plugins-plus</code>。</p><p><s>Yurchiu 已经很努力地尽量适配这个插件了哦！</s></p><details><summary>点击查看</summary><h4>行内文本样式 text</h4>1. 带<u>下划线</u>的文本；2. 带<emp>着重号</emp>的文本；3. 带<wavy>波浪线</wavy>的文本；4. 带<del>删除线</del>的文本；5. 键盘样式的文本 <kbd>command</kbd> + <kbd>D</kbd>；6. 密码样式的文本：<psw>这里没有验证码</psw>。<h4>行内文本 span</h4><strong>彩色文字</strong><p>在一段话中方便插入各种颜色的标签，包括：<span class='p red'>红色</span>、<span class='p yellow'>黄色</span>、<span class='p green'>绿色</span>、<span class='p cyan'>青色</span>、<span class='p blue'>蓝色</span>、<span class='p gray'>灰色</span>。</p><p><strong>超大号文字</strong></p><span class='p center logo large'>Cutie</span><span class='p center small'>A terrible Theme for Hexo</span><h4>段落文本 p</h4><strong>彩色文字</strong><p>在一段话中方便插入各种颜色的标签，包括：</p><p class='p red'>红色</p><p class='p yellow'>黄色</p><p class='p green'>绿色</p><p class='p cyan'>青色</p><p class='p blue'>蓝色</p><p class='p gray'>灰色</p><strong>超大号文字</strong><p class='p center logo large'>Cutie</p><p class='p center small'>A terrible Theme for Hexo</p><h4>复选列表 checkbox</h4><div class='checkbox'><input type="checkbox" />            <p>纯文本测试</p>            </div><div class='checkbox checked'><input type="checkbox" checked="checked"/>            <p>支持简单的 <a href="https://guides.github.com/features/mastering-markdown/">markdown</a> 语法</p>            </div><div class='checkbox red'><input type="checkbox" />            <p>支持自定义颜色</p>            </div><div class='checkbox green checked'><input type="checkbox" checked="checked"/>            <p>绿色 + 默认选中</p>            </div><div class='checkbox yellow checked'><input type="checkbox" checked="checked"/>            <p>黄色 + 默认选中</p>            </div><div class='checkbox cyan checked'><input type="checkbox" checked="checked"/>            <p>青色 + 默认选中</p>            </div><div class='checkbox blue checked'><input type="checkbox" checked="checked"/>            <p>蓝色 + 默认选中</p>            </div><div class='checkbox plus green checked'><input type="checkbox" checked="checked"/>            <p>增加</p>            </div><div class='checkbox minus yellow checked'><input type="checkbox" checked="checked"/>            <p>减少</p>            </div><div class='checkbox times red checked'><input type="checkbox" checked="checked"/>            <p>叉</p>            </div><h4>折叠框 folding</h4><details class="folding-tag" ><summary> 查看图片测试 </summary>              <div class='content'>              <p><img src="https://cdn.jsdelivr.net/gh/volantis-x/cdn-wallpaper/abstract/41F215B9-261F-48B4-80B5-4E86E165259E.jpeg" alt="" /></p>              </div>            </details><details class="folding-tag" cyan open><summary> 查看默认打开的折叠框 </summary>              <div class='content'>              <p>这是一个默认打开的折叠框。</p>              </div>            </details><details class="folding-tag" green><summary> 查看代码测试 </summary>              <div class='content'>              <p>貌似内部的代码不能正确换行。</p><pre class="highlight"><code class="javascript">posts.<span class="hljs-property">data</span> = posts.<span class="hljs-property">data</span>.<span class="hljs-title function_">sort</span>(<span class="hljs-keyword">function</span>(<span class="hljs-params">a, b</span>) {      <span class="hljs-keyword">if</span>(a.<span class="hljs-property">top</span> &amp;&amp; b.<span class="hljs-property">top</span>) {          <span class="hljs-keyword">if</span>(a.<span class="hljs-property">top</span> == b.<span class="hljs-property">top</span>) <span class="hljs-keyword">return</span> b.<span class="hljs-property">date</span> - a.<span class="hljs-property">date</span>;          <span class="hljs-keyword">else</span> <span class="hljs-keyword">return</span> b.<span class="hljs-property">top</span> - a.<span class="hljs-property">top</span>;      }      <span class="hljs-keyword">else</span> <span class="hljs-keyword">if</span>(a.<span class="hljs-property">top</span> &amp;&amp; !b.<span class="hljs-property">top</span>) {          <span class="hljs-keyword">return</span> -<span class="hljs-number">1</span>;      }      <span class="hljs-keyword">else</span> <span class="hljs-keyword">if</span>(!a.<span class="hljs-property">top</span> &amp;&amp; b.<span class="hljs-property">top</span>) {          <span class="hljs-keyword">return</span> <span class="hljs-number">1</span>;      }      <span class="hljs-keyword">else</span> <span class="hljs-keyword">return</span> b.<span class="hljs-property">date</span> - a.<span class="hljs-property">date</span>;});</code></pre>              </div>            </details><details class="folding-tag" yellow><summary> 查看列表测试 </summary>              <div class='content'>              <ul><li>haha</li><li>hehe</li></ul>              </div>            </details><details class="folding-tag" red><summary> 查看嵌套测试 </summary>              <div class='content'>              <details class="folding-tag" blue><summary> 查看嵌套测试2 </summary>              <div class='content'>              <details class="folding-tag" ><summary> 查看嵌套测试3 </summary>              <div class='content'>              <p>hahaha <span><img src='https://cdn.jsdelivr.net/gh/volantis-x/cdn-emoji/tieba/%E6%BB%91%E7%A8%BD.png' style='height:24px'></span></p>              </div>            </details>              </div>            </details>              </div>            </details><h4>链接卡片 link</h4><div class="tag link"><a class="link-card" title="Yurchiu" href="https://yz-hs.github.io/"><div class="left"><img src="/img/new-avatar.jpg"/></div><div class="right"><p class="text">Yurchiu</p><p class="url">https://yz-hs.github.io/</p></div></a></div><h4>进度条 progress</h4>文字不能太长。<div class="progress"><div class="progress-bar-animated progress-bar progress-bar-striped bg-yellow"  style="width: 30%" aria-valuenow="30" aria-valuemin="0" aria-valuemax="100"><p>进度条样式预览</p></div></div><div class="progress"><div class="progress-bar-animated progress-bar progress-bar-striped bg-green"  style="width: 50%" aria-valuenow="50" aria-valuemin="0" aria-valuemax="100"><p>进度条样式预览</p></div></div><div class="progress"><div class="progress-bar-animated progress-bar progress-bar-striped bg-cyan"  style="width: 70%" aria-valuenow="70" aria-valuemin="0" aria-valuemax="100"><p>进度条样式预览</p></div></div><div class="progress"><div class="progress-bar-animated progress-bar progress-bar-striped bg-blue"  style="width: 90%" aria-valuenow="90" aria-valuemin="0" aria-valuemax="100"><p>进度条样式预览</p></div></div><div class="progress"><div class="progress-bar-animated progress-bar progress-bar-striped bg-gray"  style="width: 100%" aria-valuenow="100" aria-valuemin="0" aria-valuemax="100"><p>进度条样式预览</p></div></div><h4>诗词标签 poem</h4><div class='poem'><div class='poem-title'>水调歌头</div><div class='poem-author'>苏轼</div><p>丙辰中秋，欢饮达旦，大醉，作此篇，兼怀子由。<br />明月几时有？把酒问青天。<br />不知天上宫阙，今夕是何年？<br />我欲乘风归去，又恐琼楼玉宇，高处不胜寒。<br />起舞弄清影，何似在人间？</p><p>转朱阁，低绮户，照无眠。<br />不应有恨，何事长向别时圆？<br />人有悲欢离合，月有阴晴圆缺，此事古难全。<br />但愿人长久，千里共婵娟。</p></div><h4>气泡注释 bubble</h4>最近我学到了不少新玩意儿（虽然对很多大佬来说这些已经是旧技术了），比如CSS的<span class="bubble-content">兄弟相邻选择器</span><span class="bubble-notation"><span class="bubble-item" style="background-color:#71a4e3;">例如 h1 + p {margin-top:50px;}</span></span>，<span class="bubble-content">Flex 布局</span><span class="bubble-notation"><span class="bubble-item" style="background-color:#ec5830;">Flex 是 Flexible Box 的缩写，意为弹性布局"，用来为盒状模型提供最大的灵活性"</span></span>，<span class="bubble-content">transform 变换</span><span class="bubble-notation"><span class="bubble-item" style="background-color:#1db675;">transform 属性向元素应用 2D 或 3D 转换。该属性允许我们对元素进行旋转、缩放、移动或倾斜。</span></span>，animation 的<span class="bubble-content">贝塞尔速度曲线</span><span class="bubble-notation"><span class="bubble-item" style="background-color:#de4489;">贝塞尔曲线(Bézier curve)，又称贝兹曲线或贝济埃曲线，是应用于二维图形应用程序的数学曲线。一般的矢量图形软件通过它来精确画出曲线，贝兹曲线由线段与节点组成，节点是可拖动的支点，线段像可伸缩的皮筋</span></span>写法，还有今天刚看到的 <span class="bubble-content">clip-path</span><span class="bubble-notation"><span class="bubble-item" style="background-color:#868fd7;">clip-path 属性使用裁剪方式创建元素的可显示区域。区域内的部分显示，区域外的隐藏。</span></span> 属性。这些对我来说很新颖的概念狠狠的冲击着我以前积累起来的设计思路。</details><h2 id="附注"><a class="markdownIt-Anchor" href="#附注"></a> 附注</h2><ol><li>推荐在文件 <code>path-to-your-blog/_config.yml</code> 中将 per_page 设置为 0。</li><li>为防止喧宾夺主，本主题不再提供博客美化 js。</li><li>如果一个页面中使用多个“标签页”，应使用 id 标注。例：</li></ol><div class="tab-page" id="tab1"> <div class="tabTitle">  <ul>    <li class="current">测试</li>    <li>文字</li>    <li>留空</li>    <li>嵌套</li>  </ul> </div> <div class="tabContent">  <div>测试测试测试测试测试测试测试测试测试测试</div>  <div class="hide">You AK IOI!You AK IOI!You AK IOI!You AK IOI!You AK IOI!You AK IOI!You AK IOI!You AK IOI!You AK IOI!You AK IOI!You AK IOI!You AK IOI!</div>  <div class="hide"></div>  <div class="hide">  <div class="tab-page" id="tab2"> <div class="tabTitle">  <ul>    <li class="current">可以</li>    <li>再次嵌套</li>  </ul> </div> <div class="tabContent">  <div>标签页可以进行嵌套。</div>  <div class="hide">防止占用篇幅就不再嵌套了（）</div> </div></div></div> </div></div><script>    $(function(){      var ali = $('#tab1>.tabTitle>ul>li');      var aDiv = $('#tab1>.tabContent>div');      var timeId = null;      ali.click(function(){        var _this = $(this);        timeId = setTimeout(function(){          _this.addClass('current').siblings().removeClass('current');          var index = _this.index();          aDiv.eq(index).show().siblings().hide();        });      });  });  $(function(){      var ali = $('#tab2>.tabTitle>ul>li');      var aDiv = $('#tab2>.tabContent>div');      var timeId = null;      ali.click(function(){        var _this = $(this);        timeId = setTimeout(function(){          _this.addClass('current').siblings().removeClass('current');          var index = _this.index();          aDiv.eq(index).show().siblings().hide();        });      });  });</script>]]></content>
      
      
      <categories>
          
          <category> Cutie </category>
          
      </categories>
      
      
        <tags>
            
            <tag> Hexo </tag>
            
            <tag> Cutie </tag>
            
        </tags>
      
    </entry>
    
    
    
    <entry>
      <title>Hexo 博客搭建</title>
      <link href="/2019/10/03/Hexo%20%E5%8D%9A%E5%AE%A2%E6%90%AD%E5%BB%BA/"/>
      <url>/2019/10/03/Hexo%20%E5%8D%9A%E5%AE%A2%E6%90%AD%E5%BB%BA/</url>
      
        <content type="html"><![CDATA[<h1 id="1-安装-git-nodejs"><a class="markdownIt-Anchor" href="#1-安装-git-nodejs"></a> 1. 安装 git &amp; Node.js</h1><p>请自行下载。</p><p><a href="https://gitforwindows.org/">https://gitforwindows.org/</a></p><p><a href="https://nodejs.org/en/download/">https://nodejs.org/en/download/</a></p><h1 id="2-安装-hexo"><a class="markdownIt-Anchor" href="#2-安装-hexo"></a> 2. 安装 Hexo</h1><p>先创建一个文件夹 blog，在这个文件夹下右键打开 git bash。如果 npm 不好用，使用 yarn。</p><pre class="highlight"><code class="bash">npm install -g hexo-cli</code></pre><p>初始化 Hexo。</p><pre class="highlight"><code class="bash">hexo initnpm install</code></pre><p>打开 Hexo 的服务，</p><pre class="highlight"><code class="bash">hexo ghexo server</code></pre><p>在浏览器输入 <code>localhost:4000</code> 就可以看到你生成的博客了。</p><p>Ctrl+C 关闭。</p><h1 id="3-github-创建个人仓库"><a class="markdownIt-Anchor" href="#3-github-创建个人仓库"></a> 3. GitHub 创建个人仓库</h1><p>你先要有一个 GitHub 账户，去<a href="https://github.com/">注册一个吧</a>。</p><p>创建一个和你用户名相同的仓库，<strong>后面加 <code>.github.io</code></strong></p><p><img src="https://cdn.jsdelivr.net/gh/yz-hs/PicGo/%E6%90%ADhexo%20(1).png" alt="" /></p><p><img src="https://cdn.jsdelivr.net/gh/yz-hs/PicGo/%E6%90%ADhexo%20(2).png" alt="" /></p><p>点击 create repository。</p><h1 id="4-生成-ssh-添加到-github"><a class="markdownIt-Anchor" href="#4-生成-ssh-添加到-github"></a> 4. 生成 SSH 添加到 GitHub</h1><p>回到 git bash 中，</p><pre class="highlight"><code class="bash">git config --global user.name <span class="hljs-string">&quot;yourname&quot;</span>git config --global user.email <span class="hljs-string">&quot;youremail&quot;</span></code></pre><p>这里的 yourname 输入你的 GitHub 用户名，youremail 输入你 GitHub 的邮箱。</p><p>可以用以下两条，检查一下你有没有输对。</p><pre class="highlight"><code class="bash">git config user.namegit config user.email</code></pre><p>然后创建 SSH,一路回车。</p><pre class="highlight"><code class="bash">ssh-keygen -t rsa -C <span class="hljs-string">&quot;youremail&quot;</span></code></pre><p>这个时候它会告诉你已经生成了 .ssh 的文件夹。在你的电脑中找到这个文件夹。</p><p>ssh，简单来讲，就是一个秘钥，其中，id_rsa 是你这台电脑的私人秘钥，不能给别人看的，id_rsa.pub 是公共秘钥，可以随便给别人看。把这个公钥放在 GitHub 上，这样当你链接 GitHub 自己的账户时，它就会根据公钥匹配你的私钥，当能够相互匹配时，才能够顺利的通过 git 上传你的文件到 GitHub 上。</p><p>而后在 GitHub 的 setting 中，找到 SSH keys 的设置选项，点击 New SSH key 把你的 id_rsa.pub 里面的信息复制进去。</p><pre class="highlight"><code class="bash">ssh -T git@github.com</code></pre><p>如果出现让你写 yes 或 no,写 yes。</p><h1 id="5-将-hexo-部署到-github"><a class="markdownIt-Anchor" href="#5-将-hexo-部署到-github"></a> 5. 将 Hexo 部署到 GitHub</h1><p>这一步，我们就可以将 Hexo 和 GitHub 关联起来，也就是将 Hexo 生成的文章部署到 GitHub 上，打开站点配置文件 <code>_config.yml</code>，翻到最后，修改为 YourName 就是你的 GitHub 账户</p><pre class="highlight"><code class="yml"><span class="hljs-attr">deploy:</span>  <span class="hljs-attr">type:</span> <span class="hljs-string">git</span>  <span class="hljs-attr">repository:</span> <span class="hljs-string">git@github.com:YourName/YourName.github.io.git</span>  <span class="hljs-attr">branch:</span> <span class="hljs-string">master</span></code></pre><p>安装 deploy-git，也就是部署的命令,这样你才能用命令部署到 GitHub。</p><pre class="highlight"><code class="bash">npm install hexo-deployer-git --save</code></pre><p>然后</p><pre class="highlight"><code class="bash">hexo cleanhexo generatehexo deploy</code></pre><p>其中 hexo clean 清除了你之前生成的东西，也可以不加。 hexo generate 顾名思义，生成静态文章，可以用 hexo g 缩写；hexo deploy 部署文章，可以用 hexo d 缩写</p><p>出现 <code>INFO  Deploy done: git</code> 就说明部署成功了，过一会儿就可以在 <code>http://yourname.github.io</code> 这个网站看到你的博客了。</p><h1 id="6-博客备份"><a class="markdownIt-Anchor" href="#6-博客备份"></a> 6. 博客备份</h1><p>为了保证我们写的文章不丢失、快速迁移博客，都需要备份我们的 blog。</p><ol><li><p>博客根目录，执行 <code>git init</code> 创建 git 仓库。</p></li><li><p>在 github 创建仓库并和本地仓库建立联系。</p></li><li><p>在<code>~/.bashrc</code> 文件中添加以下内容：</p><pre class="highlight"><code class="">alias hs='hexo clean &amp;&amp; hexo g &amp;&amp; hexo s'alias hd='hexo clean &amp;&amp; hexo g &amp;&amp; hexo d &amp;&amp; git add . &amp;&amp; git commit -m &quot;update&quot; &amp;&amp; git push -f'</code></pre></li></ol><p>这样，执行<code>hs</code>即可启动本地服务；执行 <code>hd</code> 进行部署博客时，就一同将博客进行备份了。</p><h1 id="7-参考"><a class="markdownIt-Anchor" href="#7-参考"></a> 7. 参考</h1><p><a href="https://zhuanlan.zhihu.com/p/44213627">https://zhuanlan.zhihu.com/p/44213627</a></p><p>接下来就要你自己配置了。</p>]]></content>
      
      
      <categories>
          
          <category> Hexo </category>
          
      </categories>
      
      
        <tags>
            
            <tag> Hexo </tag>
            
        </tags>
      
    </entry>
    
    
    
    <entry>
      <title>欢迎</title>
      <link href="/2019/08/19/%E6%AC%A2%E8%BF%8E/"/>
      <url>/2019/08/19/%E6%AC%A2%E8%BF%8E/</url>
      
        <content type="html"><![CDATA[<p>欢迎来到 Yurchiu 的博客！希望我们能相互交流，共同进步～</p><span id="more"></span><!-- https://cdn.jsdelivr.net/gh/yz-hs/PicGo/newyear-welcome.jpg --><h1 id="欢迎-dalao-来访"><a class="markdownIt-Anchor" href="#欢迎-dalao-来访"></a> 欢迎 Dalao 来访</h1><h2 id="why-blog"><a class="markdownIt-Anchor" href="#why-blog"></a> Why Blog</h2><div class="alert alert-info">Why Blog“三段论”来自 yelog，并非原创。</div>喜欢写 Blog 的人，会经历三个阶段。<blockquote><p>第一阶段，刚接触 Blog，觉得很新鲜，试着选择一个免费空间来写。</p></blockquote><blockquote><p>第二阶段，发现免费空间限制太多，就自己购买域名和空间，搭建独立博客。</p></blockquote><blockquote><p>第三阶段，觉得独立博客的管理太麻烦，最好在保留控制权的前提下，让别人来管，自己只负责写文章。</p></blockquote><p>我们每个人的在网络上产生的数据越来越多，这些信息是我们在互联网上存在过的痕迹，值得我们认真对待。但是它们被分散分布在各个网站上。很多时候我们很难将它们聚合在一起，而且各个网站的信息排布方式也没有办法自由控制，所以我们需要一个可以由自己主宰的空间——博客。</p><p>通过博客，我们可以记录自己的生活和成长的轨迹。它不像 Twitter 那样碎片化，也不像 Facebook 那样关系化，它是私人的空间。</p><h1 id="网站指南"><a class="markdownIt-Anchor" href="#网站指南"></a> 网站指南</h1><h2 id="侧边栏"><a class="markdownIt-Anchor" href="#侧边栏"></a> 侧边栏</h2><ul><li>说说：一些简短的话，不至于发新文章，以说说形式发布。</li><li>留言：有什么想法、建议、吐槽、Bug 反馈，可以留言！</li><li>链接：一些友链，以及有用的网站。</li></ul><h2 id="已知-bug"><a class="markdownIt-Anchor" href="#已知-bug"></a> 已知 Bug</h2><ul><li>本主题或多或少有一些无法解决的 Bug。</li><li>题解的 CODE 会不可避免地被 HACK！Dalao 们若发现错误 / 不明确的地方请提出您宝贵的意见。</li></ul><h2 id="其他"><a class="markdownIt-Anchor" href="#其他"></a> 其他</h2><p>本站域名不再更换（AFO 了），请直接访问 <a href="https://Yurchiu.github.io">https://Yurchiu.github.io</a>。</p><p>更换记录：<code>yz-hs.github.io</code> -&gt; <code>orzyz.tk</code> -&gt; <code>yz-hs.tk</code> -&gt; <code>cieu.tk</code> -&gt; <code>yz-hs.github.io</code> -&gt; <code>Yurchiu.github.io</code>。</p>]]></content>
      
      
      
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