\u672c\u6587\u539f\u4f5c\u8005\uff1a[JustPureH2O](https:\/\/justpureh2o.cn\/about\/)<\/p>\n\n\n\n
\u539f\u6587\u8fde\u63a5\uff1ahttps:\/\/justpureh2o.cn\/articles\/61430\/<\/p>\n\n\n\n
# \u5706\u9525\u66f2\u7ebf \u5e38\u7528\u4e8c\u7ea7\u7ed3\u8bba\u9644\u8bc1\u660e\u8fc7\u7a0b<\/strong><\/p>\n\n\n\n ## \u524d\u8a00<\/strong><\/p>\n\n\n\n \u505a\u8fd9\u7bc7\u6587\u7ae0\u6700\u521d\u7684\u7f18\u7531\uff0c\u4f3c\u4e4e\u4e5f\u65e9\u5df2\u5fd8\u5374\u4e86\uff0c\u5927\u62b5\u662f\u671f\u4e2d\u65f6\u62cd\u62cd\u8111\u5b50\u751f\u51fa\u7684\u4e3b\u610f\uff0c\u7136\u7ec8\u7adf\u65e0\u6cd5\u518d\u8003\u7a76\u4e86\u2026\u2026\u7136\u800c\u65e2\u5df2\u5165\u5e74\uff0c\u6211\u60f3\u81ea\u5df1\u8bb8\u662f\u6709\u4e9b\u5bc2\u5bde\u800c\u65e0\u6240\u4e8b\u4e8b\u4e86\u3002\u6070\u9022 A \u541b\u9080\u6211\u505a\u4e00\u7bc7\u5706\u9525\u66f2\u7ebf\u7684\u6587\u7ae0\uff0c\u4ed6\u4eff\u4f5b\u6709\u70b9\u8c11\u6211\u7684\u610f\u601d\uff0c\u6b32\u4ee5\u6b64\u6307\u4ee3\u65b9\u624d\u8fc7\u53bb\u7684\u4ee4\u4eba\u60b2\u54c0\u7684\u6570\u5b66\u8003\u9898\uff0c\u6211\u4e3e\u8d77\u624b\u673a\u53ea\u662f\u8bf4\uff1a<\/p>\n\n\n\n \u201c\u5047\u5982\u4e00\u9053\u5706\u9525\u66f2\u7ebf\u586b\u7a7a\u538b\u8f74\uff0c\u5b83\u662f\u6ca1\u6709\u5e38\u6570\u4e14\u4e07\u96be\u7b97\u51fa\u7684\uff0c\u8003\u573a\u4e0a\u6709\u8bb8\u591a\u6ca1\u80cc\u4e8c\u7ea7\u7ed3\u8bba\u7684\u9ad8\u4e2d\u751f\uff0c\u89c1\u5230\u5c31\u8df3\u8fc7\u4e86\uff0c\u7136\u800c\u662f\u820d\u5c0f\u4fdd\u5927\uff0c\u5e76\u4e0d\u611f\u5230\u6302\u79d1\u7684\u60b2\u54c0\u3002\u73b0\u5728\u4f60\u6559\u4ed6\u4eec\u80cc\u4e8c\u7ea7\u7ed3\u8bba\uff0c\u8bf4\u52a8\u4e86\u60f3\u51b2\u9ad8\u5206\u7684\u51e0\u4eba\uff0c\u4f7f\u8fd9\u4e0d\u5e78\u7684\u5c11\u6570\u8005\u6765\u53d7\u77e5\u6653\u7ed3\u8bba\u800c\u4ecd\u89e3\u4e0d\u51fa\u7684\u53ef\u80fd\u6302\u79d1\u7684\u82e6\u695a\uff0c\u4f60\u5012\u4ee5\u4e3a\u5bf9\u5f97\u8d77\u4ed6\u4eec\u4e48\uff1f\u201d<\/p>\n\n\n\n \u201c\u7136\u800c\u51e0\u4e2a\u4eba\u65e2\u7136\u80cc\u4e86\uff0c\u4f60\u4e0d\u80fd\u8bf4\u51b3\u6ca1\u6709\u89e3\u51fa\u8fd9\u4e2a\u9898\u7684\u5e0c\u671b\u3002\u201d<\/p>\n\n\n\n \u662f\u7684\uff0c\u6211\u867d\u7136\u81ea\u6709\u6211\u7684\u786e\u4fe1\uff0c\u7136\u800c\u8bf4\u5230\u5e0c\u671b\uff0c\u5374\u662f\u4e0d\u80fd\u62b9\u6740\u7684\u3002\u56e0\u4e3a\u5e0c\u671b\u662f\u5728\u4e8e\u5c06\u6765\uff0c\u51b3\u4e0d\u80fd\u4ee5\u6211\u4e4b\u5fc5\u65e0\u7684\u8bc1\u660e\uff0c\u6765\u6298\u670d\u4e86\u4ed6\u4e4b\u6240\u8c13\u53ef\u6709\uff0c\u4e8e\u662f\u6211\u7ec8\u4e8e\u7b54\u5e94\u4ed6\u4e5f\u505a\u5706\u9525\u66f2\u7ebf\u7684\u6587\u7ae0\u4e86\u3002<\/p>\n\n\n\n {\n\n\n\n \n\n\n\n ## \u692d\u5706 \u57fa\u7840\u4e8c\u7ea7\u7ed3\u8bba<\/strong><\/p>\n\n\n\n \u82e5\u65e0\u7279\u6b8a\u8bf4\u660e\uff0c\u692d\u5706\u6807\u51c6\u65b9\u7a0b $E: \\frac{x^2}{a^2}+\\frac{y^2}{b^2}=1$ \u5747\u6ee1\u8db3 $a>b>0$\uff0c\u7126\u70b9\u5728 $x$ \u8f74\u4e0a\u3002\u4e14\u82e5\u65e0\u7279\u6b8a\u6307\u660e\uff0c\u201c\u692d\u5706 $E$\u201d\u5747\u6307\u4e0a\u8ff0\u7684\u692d\u5706 $E: \\frac{x^2}{a^2}+\\frac{y^2}{b^2}=1$\u3002<\/p>\n\n\n\n ### \u901a\u5f84<\/strong><\/p>\n\n\n\n > \u5728\u692d\u5706 $E$ \u4e2d\uff0c\u4e0e\u7126\u70b9\u6240\u5728\u8f74\u5782\u76f4\u7684\u7126\u70b9\u5f26\u88ab\u692d\u5706\u622a\u5f97\u7ebf\u6bb5\u7684\u957f\u79f0\u4f5c\u5176\u901a\u5f84\u3002\u692d\u5706\u7684\u901a\u5f84\u957f $d=\\frac{2b^2}{a}$\u3002<\/p>\n\n\n\n **\u8bc4\u6790**<\/strong>\uff1a\u76f4\u63a5\u5c06\u6a2a\u5750\u6807 $\\pm c$ \u4ee3\u5165\u5373\u53ef\u89e3\u5f97\u7eb5\u5750\u6807\uff0c\u901a\u5f84\u957f\u4e3a\u7eb5\u5750\u6807\u7edd\u5bf9\u503c\u7684\u4e8c\u500d\u3002<\/p>\n\n\n\n **\u8bc1\u660e**<\/strong>\uff1a<\/p>\n\n\n\n \u6a2a\u5750\u6807 $c$ \u4ee3\u5165\u692d\u5706\u89e3\u6790\u5f0f\uff1a<\/p>\n\n\n\n $$<\/p>\n\n\n\n \\frac{c^2}{a^2}+\\frac{y^2}{b^2}=1<\/p>\n\n\n\n $$<\/p>\n\n\n\n \u5f97\u5230 $y^2=b^2-b^2e^2$\u3002<\/p>\n\n\n\n \u56e0\u4e3a\u5728\u692d\u5706\u4e2d\u6709 $a^2=b^2+c^2$\uff0c\u56e0\u6b64 $c^2=a^2-b^2$\uff0c\u692d\u5706\u79bb\u5fc3\u7387\u8fd8\u53ef\u4ee5\u8868\u793a\u6210 $e=\\frac{c}{a}=\\sqrt{\\frac{c^2}{a^2}}=\\sqrt{\\frac{a^2-b^2}{a^2}}=\\sqrt{1-\\frac{b^2}{a^2}}$\u3002\u4ee3\u5165\u5f97\uff1a<\/p>\n\n\n\n $$<\/p>\n\n\n\n \\begin{aligned}<\/p>\n\n\n\n y^2&=b^2-e^2b^2<\/p>\n\n\n\n \\\\&=b^2-(1-\\frac{b^2}{a^2})b^2<\/p>\n\n\n\n \\\\&=\\frac{b^4}{a^2}<\/p>\n\n\n\n \\\\y&=\\pm\\frac{b^2}{a}<\/p>\n\n\n\n \\end{aligned}<\/p>\n\n\n\n $$<\/p>\n\n\n\n \u6b64\u65f6 $d=2|y|=\\frac{2b^2}{a}$\u3002<\/p>\n\n\n\n \u8bc1\u6bd5\u3002<\/p>\n\n\n\n ### \u5706\u5468\u5b9a\u7406<\/strong><\/p>\n\n\n\n > \u5728\u692d\u5706 $E$ \u4e2d\uff0c$A,B$ \u662f\u692d\u5706\u4e0a\u5173\u4e8e\u539f\u70b9\u5bf9\u79f0\u7684\u4e24\u70b9\uff0c$M$ \u662f\u692d\u5706\u4e0a\u5f02\u4e8e $A,B$ \u7684\u4e00\u70b9\u3002\u90a3\u4e48\u76f4\u7ebf $AM,BM$ \u7684\u659c\u7387\u4e4b\u79ef\u4e3a $-\\frac{b^2}{a^2}$\u3002<\/p>\n\n\n\n **\u8bc4\u6790**<\/strong>\uff1a\u4e5f\u4e0d\u9700\u8981\u4ec0\u4e48\u7279\u6b8a\u7684\u6280\u5de7\uff0c\u5c31\u662f\u8bbe\u70b9\u786c\u7b97\u3002\u8fd9\u4e2a\u7ed3\u8bba\u662f\u5fc5\u80cc\u7684\u7ecf\u5178\u4e8c\u7ea7\u7ed3\u8bba\u4e4b\u4e00\u3002<\/p>\n\n\n\n **\u8bc1\u660e**<\/strong>\uff1a<\/p>\n\n\n\n \u8bbe $A(x_1,y_1)~B(-x_1,-y_1)~M(x_2,y_2)$\u3002\u90a3\u4e48\u4e24\u76f4\u7ebf\u659c\u7387\u4e4b\u79ef\u53ef\u4ee5\u8868\u793a\u4e3a $k_{AM}k_{BM}$\uff0c\u5373\uff1a<\/p>\n\n\n\n $$<\/p>\n\n\n\n \\begin{aligned}<\/p>\n\n\n\n k_{AM}\\cdot k_{BM}&=\\frac{y_2-y_1}{x_2-x_1}\\times\\frac{y_2+y_1}{x_2+x_1}<\/p>\n\n\n\n \\\\&=\\frac{y_2^2-y_1^2}{x_2^2-x_1^2}<\/p>\n\n\n\n \\end{aligned}<\/p>\n\n\n\n $$<\/p>\n\n\n\n \u4e09\u70b9\u90fd\u5728\u692d\u5706\u4e0a\uff0c\u4ee3\u5165\u692d\u5706\u89e3\u6790\u5f0f\u5f97\u5173\u7cfb\u5f0f\uff1a<\/p>\n\n\n\n $$<\/p>\n\n\n\n \\begin{cases}<\/p>\n\n\n\n \\dfrac{x_1^2}{a^2}+\\dfrac{y_1^2}{b^2}=1<\/p>\n\n\n\n \\\\\\qquad<\/p>\n\n\n\n \\\\\\dfrac{x_2^2}{a^2}+\\dfrac{y_2^2}{b^2}=1<\/p>\n\n\n\n \\end{cases}\\rightarrow\\begin{cases}<\/p>\n\n\n\n y_1^2=b^2-\\dfrac{b^2x_1^2}{a^2}<\/p>\n\n\n\n \\\\\\qquad<\/p>\n\n\n\n \\\\y_2^2=b^2-\\dfrac{b^2x_2^2}{a^2}<\/p>\n\n\n\n \\end{cases}<\/p>\n\n\n\n $$<\/p>\n\n\n\n \u5c06 $y_1^2$ \u548c $y_2^2$ \u7528 $x_1^2,x_2^2$ \u8868\u793a\u51fa\u6765\uff1a<\/p>\n\n\n\n $$<\/p>\n\n\n\n \\begin{aligned}<\/p>\n\n\n\n &=\\dfrac{\\frac{b^2}{a^2}(x_1^2-x_2^2)}{x_2^2-x_1^2}<\/p>\n\n\n\n \\\\&=-\\frac{b^2}{a^2}<\/p>\n\n\n\n \\end{aligned}<\/p>\n\n\n\n $$<\/p>\n\n\n\n \u8bc1\u6bd5\u3002<\/p>\n\n\n\n **\u62d3\u5c55\u53d8\u5f62**<\/strong>\uff1a\u6211\u4eec\u9ed8\u8ba4\u692d\u5706\u7684\u7126\u70b9\u4f4d\u4e8e $x$ \u8f74\uff0c\u90a3\u4e07\u4e00\u7126\u70b9\u5728 $y$ \u8f74\u4e0a\u5462\uff1f\u9996\u5148\u6211\u4eec\u9700\u8981\u4fdd\u8bc1\u8f83\u5927\u7684\u5206\u6bcd\u4e3a $a$\uff0c\u8f83\u5c0f\u7684\u4e3a $b$\uff0c\u4f8b\u5982 $\\frac{y^2}{9}+\\frac{x^2}{4}=1$\uff0c\u6b64\u65f6 $a^2=9,b^2=4$\uff0c\u73b0\u5728\u7684\u4e24\u76f4\u7ebf\u659c\u7387\u4e4b\u79ef\u4e3a $-\\frac{9}{4}$\uff0c\u5373 $-\\frac{a^2}{b^2}$\uff0c\u5206\u5b50\u5206\u6bcd\u8c03\u6362\u4e86\uff01\u505a\u9898\u65f6\u4e00\u5b9a\u8981\u6ce8\u610f\uff0c\u8bc1\u660e\u65b9\u6cd5\u540c\u4e0a\u3002<\/p>\n\n\n\n ### \u5e7f\u4e49\u5782\u5f84\u5b9a\u7406\/\u4e2d\u70b9\u5f26\u516c\u5f0f<\/strong><\/p>\n\n\n\n > \u5728\u692d\u5706 $E$ \u4e2d\uff0c$A,B$ \u4e3a\u692d\u5706\u4e0a\u4e24\u70b9\uff0c$M$ \u4e3a\u5f26 $AB$ \u7684\u4e2d\u70b9\uff0c\u90a3\u4e48\u76f4\u7ebf $OM$ \u4e0e\u76f4\u7ebf $AB$ \u7684\u659c\u7387\u4e4b\u79ef\u4e3a $-\\frac{b^2}{a^2}$\u3002<\/p>\n\n\n\n **\u8bc4\u6790**<\/strong>\uff1a\u5904\u7406\u4e2d\u70b9\u7684\u65b9\u6cd5\u4e00\u5171\u6709\u4e24\u4e2a\u2014\u2014\u5e38\u89c4\u8054\u7acb\u6cd5\u548c\u70b9\u5dee\u6cd5\u3002\u6b64\u5904\u6211\u4eec\u4f7f\u7528\u7b2c\u4e00\u79cd\uff0c\u56e0\u4e3a\u97e6\u8fbe\u5b9a\u7406\u53ef\u4ee5\u5f88\u8f7b\u677e\u7684\u8868\u793a\u51fa\u4e2d\u70b9\u7684\u5750\u6807\uff1b\u540c\u65f6\u8bbe\u51fa\u76f4\u7ebf $AB$ \u4ee3\u8868\u6211\u4eec\u53ef\u4ee5\u53ea\u7528\u4e00\u4e2a $k$ \u8868\u793a\u5176\u659c\u7387\uff0c\u7531\u4e8e $OM$ \u8fc7\u539f\u70b9\uff0c\u8868\u793a\u5b83\u7684\u659c\u7387\u4e5f\u662f\u5bb9\u6613\u7684\u3002\u90a3\u6211\u4eec\u5c31\u5f00\u59cb\u5427\u3002<\/p>\n\n\n\n **\u8bc1\u660e**<\/strong>\uff1a<\/p>\n\n\n\n \u8bbe\u76f4\u7ebf $AB:y=kx+m$\u3002\u659c\u7387\u4e0d\u5b58\u5728\u65f6\u65e0\u610f\u4e49\uff0c\u6545\u659c\u7387\u4e00\u5b9a\u5b58\u5728\u3002\u8054\u7acb\u65b9\u7a0b\uff1a<\/p>\n\n\n\n $$<\/p>\n\n\n\n \\begin{cases}<\/p>\n\n\n\n y=kx+m<\/p>\n\n\n\n \\\\\\frac{x^2}{a^2}+\\frac{y^2}{b^2}=1<\/p>\n\n\n\n \\end{cases}<\/p>\n\n\n\n $$<\/p>\n\n\n\n \u5f97\uff1a<\/p>\n\n\n\n $$<\/p>\n\n\n\n \\begin{aligned}<\/p>\n\n\n\n b^2x^2+a^2(k^2x^2+2mkx+m^2)-a^2b^2&=0<\/p>\n\n\n\n \\\\(b^2+a^2k^2)x^2+2a^2mkx+a^2m^2-a^2b^2&=0<\/p>\n\n\n\n \\end{aligned}<\/p>\n\n\n\n $$<\/p>\n\n\n\n \u6839\u636e\u97e6\u8fbe\u5b9a\u7406\u5f97\uff1a<\/p>\n\n\n\n $$<\/p>\n\n\n\n x_1+x_2=-\\frac{2a^2mk}{b^2+a^2k^2},x_1x_2=\\frac{a^2m^2-a^2b^2}{b^2+a^2k^2}<\/p>\n\n\n\n $$<\/p>\n\n\n\n \u56e0\u6b64 $M\\left(-\\dfrac{a^2mk}{b^2+a^2k^2},\\dfrac{b^2m}{b^2+a^2k^2}\\right)$\u3002\u5f97\u5230\u659c\u7387\u4e58\u79ef\u4e3a $k\\cdot\\left(-\\dfrac{b^2m}{a^2mk}\\right)=-k\\cdot\\dfrac{b^2}{a^2k}=-\\dfrac{b^2}{a^2}$\u3002<\/p>\n\n\n\n \u8bc1\u6bd5\u3002<\/p>\n\n\n\n **\u62d3\u5c55\u53d8\u5f62**<\/strong>\uff1a\u6613\u9519\u70b9\u4e0e\u4e0a\u4e00\u4e2a\u7ed3\u8bba\u76f8\u540c\uff0c\u7126\u70b9\u6240\u5728\u5750\u6807\u8f74\u6539\u53d8\u540e\uff0c\u4e58\u79ef\u4f1a\u4ece\u539f\u5148\u7684 $-\\frac{b^2}{a^2}$ \u53d8\u6210 $-\\frac{a^2}{b^2}$\u3002<\/p>\n\n\n\n ### \u7126\u534a\u5f84\u516c\u5f0f<\/strong><\/p>\n\n\n\n > \u4ee4 $F_1,F_2$ \u4e3a\u692d\u5706 $E$ \u7684\u5de6\u53f3\u7126\u70b9\uff0c$P(x_0,y_0)$ \u4e3a\u692d\u5706\u4e0a\u4e00\u70b9\uff0c\u90a3\u4e48 $|PF_1|=a+ex_0,|PF_2|=a-ex_0$\u3002\u5176\u4e2d $e=\\frac{c}{a}$\uff0c\u5373\u692d\u5706\u7684\u79bb\u5fc3\u7387\u3002<\/p>\n\n\n\n **\u8bc4\u6790**<\/strong>\uff1a\u8fd9\u4e2a\u7ed3\u8bba\u5176\u5b9e\u5c31\u662f\u692d\u5706\u7b2c\u4e8c\u5b9a\u4e49\u7684\u53d8\u5f62\u5f0f\uff0c\u4e0d\u4fe1\u4f60\u770b\u3002<\/p>\n\n\n\n **\u8bc1\u660e**<\/strong>\uff1a<\/p>\n\n\n\n \u5148\u8bc1 $|PF_1|=a+ex_0$\u3002\u5bf9\u4e8e $F_1$ \u6765\u8bf4\uff0c\u5bf9\u5e94\u7684\u51c6\u7ebf\u4e3a\u76f4\u7ebf $x=-\\frac{a^2}{c}$\u3002\u6839\u636e\u692d\u5706\u7b2c\u4e8c\u5b9a\u4e49\u6709\uff1a<\/p>\n\n\n\n $$<\/p>\n\n\n\n \\begin{aligned}<\/p>\n\n\n\n \\dfrac{|PF_1|}{x_0+\\frac{a^2}{c}}&=e<\/p>\n\n\n\n \\\\\\dfrac{|PF_1|}{x_0+\\frac{a}{e}}&=e<\/p>\n\n\n\n \\\\|PF_1|&=ex_0+a<\/p>\n\n\n\n \\end{aligned}<\/p>\n\n\n\n $$<\/p>\n\n\n\n \u518d\u8bc1 $|PF_2|=a-ex_0$\u3002\u5176\u5b9e\u6839\u636e\u692d\u5706\u7b2c\u4e00\u5b9a\u4e49 $|PF_1|+|PF_2|=2a$ \u5373\u53ef\u63a8\u51fa\u5b83\uff0c\u4f46\u662f\u6211\u4eec\u7ee7\u7eed\u7528\u7b2c\u4e8c\u5b9a\u4e49\u63a8\u5bfc\u3002\u6b64\u65f6\u5bf9\u5e94\u7684\u51c6\u7ebf\u662f\u76f4\u7ebf $x=\\frac{a^2}{c}$\uff1a<\/p>\n\n\n\n $$<\/p>\n\n\n\n \\begin{aligned}<\/p>\n\n\n\n \\dfrac{|PF_2|}{\\frac{a}{e}-x_0}&=e<\/p>\n\n\n\n \\\\|PF_2|&=a-ex_0<\/p>\n\n\n\n \\end{aligned}<\/p>\n\n\n\n $$<\/p>\n\n\n\n \u8bc1\u6bd5\u3002<\/p>\n\n\n\n **\u62d3\u5c55\u53d8\u5f62**<\/strong>\uff1a\u7126\u70b9\u5728 $y$ \u8f74\u4e0a\u65f6\uff0c\u7ed3\u8bba\u53d8\u4e3a $|PF_1|=a+ey_0,|PF_2|=a-ey_0$\u3002<\/p>\n\n\n\n ### \u7126\u70b9\u4e09\u89d2\u5f62\u76f8\u5173<\/strong><\/p>\n\n\n\n > \u692d\u5706\u7684\u5de6\u53f3\u7126\u70b9 $F_1,F_2$ \u4e0e\u692d\u5706\u4e0a\u4e00\u70b9 $P$ \u7ec4\u6210\u7684\u4e09\u89d2\u5f62 $\\triangle PF_1F_2$ \u79f0\u4f5c\u8fd9\u4e2a\u692d\u5706\u7684\u7126\u70b9\u4e09\u89d2\u5f62\u3002<\/p>\n\n\n\n \u672c\u8282\u4e2d\u51fa\u73b0\u7684\u89d2 $\\theta$ \u82e5\u65e0\u7279\u6b8a\u8bf4\u660e\u5747\u6307\u4ee3 $\\angle F_1PF_2$\u3002<\/p>\n\n\n\n #### \u53d6\u503c\u8303\u56f4<\/strong><\/p>\n\n\n\n > \u7126\u70b9\u4e09\u89d2\u5f62 $\\triangle PF_1F_2$ \u4e2d\uff0c$|PF_1|\\in(a-c,a+c),|PF_2|\\in(a-c,a+c),|PF_1||PF_2|\\leq a^2$\u3002<\/p>\n\n\n\n **\u8bc4\u6790**<\/strong>\uff1a\u524d\u4e24\u4e2a\u975e\u5e38\u597d\u8bc1\uff0c\u4ed6\u4eec\u7406\u8bba\u4e0a\u5728 $P$ \u4e0e\u5de6\u53f3\u7aef\u70b9\u91cd\u5408\u65f6\u53d6\u5230\u6700\u503c\uff0c\u4f46\u662f\u6b64\u65f6 $P,F_1,F_2$ \u4e09\u70b9\u5171\u7ebf\uff0c\u56e0\u6b64\u4e0d\u662f\u4e09\u89d2\u5f62\uff0c\u6240\u4ee5\u662f\u5f00\u533a\u95f4\u3002\u5bf9\u4e8e\u7b2c\u4e09\u4e2a\uff0c\u4e58\u79ef\u7684\u53d6\u503c\u8303\u56f4\uff0c\u5219\u9700\u8981\u57fa\u672c\u4e0d\u7b49\u5f0f\u3002<\/p>\n\n\n\n **\u8bc1\u660e**<\/strong>\uff1a<\/p>\n\n\n\n \u6839\u636e\u57fa\u672c\u4e0d\u7b49\u5f0f\u6709\uff1a<\/p>\n\n\n\n $$<\/p>\n\n\n\n |PF_1||PF_2|\\leq\\left(\\frac{|PF_1|+|PF_2|}{2}\\right)^2=a^2<\/p>\n\n\n\n $$<\/p>\n\n\n\n \u8bc1\u6bd5\u3002<\/p>\n\n\n\n #### \u5468\u957f<\/strong><\/p>\n\n\n\n > \u7126\u70b9\u4e09\u89d2\u5f62 $\\triangle PF_1F_2$ \u7684\u5468\u957f $C_{\\triangle PF_1F_2}=|PF_1|+|PF_2|+|F_1F_2|=2a+2c$\u3002<\/p>\n\n\n\n **\u8bc4\u6790**<\/strong>\uff1a\u6839\u636e\u692d\u5706\u7684\u7b2c\u4e00\u5b9a\u4e49\u6765\u7684\uff0c$|PF_1|+|PF_2|=2a,|F_1F_2|=2c$\u3002<\/p>\n\n\n\n #### \u9762\u79ef<\/strong><\/p>\n\n\n\n > \u7126\u70b9\u4e09\u89d2\u5f62 $\\triangle PF_1F_2$ \u7684\u9762\u79ef $S_{\\triangle F_1F_2}=b^2\\tan\\frac{\\theta}{2}$\u3002<\/p>\n\n\n\n **\u8bc4\u6790**<\/strong>\uff1a\u51fa\u73b0\u89d2\u5ea6\u548c\u9762\u79ef\uff0c\u6211\u4eec\u9700\u8981\u60f3\u5230\u6b63\/\u4f59\u5f26\u5b9a\u7406\u3002\u6839\u636e\u6b63\u5f26\u5b9a\u7406\u7684\u4e09\u89d2\u5f62\u9762\u79ef\u516c\u5f0f $S=\\frac{1}{2}ab\\sin\\theta$ \u4ee5\u53ca\u4f59\u5f26\u5b9a\u7406\u7684 $c^2=a^2+b^2-2ab\\cos\\theta$\uff0c\u6211\u4eec\u53ef\u4ee5\u89e3\u51b3\u5927\u90e8\u5206\u4e0e\u8fb9\u957f\u548c\u89d2\u5ea6\u6709\u5173\u7684\u5706\u9525\u66f2\u7ebf\u8bc1\u660e\/\u6c42\u503c\u95ee\u9898\u3002<\/p>\n\n\n\n **\u8bc1\u660e**<\/strong>\uff1a<\/p>\n\n\n\n \u7531\u6b63\u5f26\u5b9a\u7406\u5f97\uff0c$S_{\\triangle PF_1F_2}=\\frac{1}{2}|PF_1||PF_2|\\sin\\theta$\u3002<\/p>\n\n\n\n \u5728 $\\triangle PF_1F_2$ \u4e2d\u8fd0\u7528\u4f59\u5f26\u5b9a\u7406\uff1a$|F_1F_2|^2=4c^2=|PF_1|^2+|PF_2|^2-2|PF_1||PF_2|\\cos\\theta$\uff0c\u5f97\u5230 $|PF_1||PF_2|=\\dfrac{|PF_1|^2+|PF_2|^2-4c^2}{2\\cos\\theta}$\u3002<\/p>\n\n\n\n \u540c\u65f6\u6839\u636e\u5b8c\u5168\u5e73\u65b9\u516c\u5f0f\uff0c$|PF_1|^2+|PF_2|^2=(|PF_1|+|PF_2|)^2-2|PF_1||PF_2|=4a^2-2|PF_1||PF_2|$\uff0c\u4ee3\u5165\u4e0a\u5f0f\u79fb\u9879\u89e3\u5f97 $|PF_1||PF_2|=\\dfrac{2(a^2-c^2)}{1+\\cos\\theta}$\uff0c\u6839\u636e\u9762\u79ef\u516c\u5f0f\u53ef\u5f97 $S=\\dfrac{(a^2-c^2)\\sin\\theta}{1+\\cos\\theta}=\\dfrac{b^2\\sin\\theta}{1+\\cos\\theta}=b^2\\tan\\dfrac{\\theta}{2}$\u3002<\/p>\n\n\n\n \u8bc1\u6bd5\u3002<\/p>\n\n\n\n **\u62d3\u5c55\u53d8\u5f62**<\/strong>\uff1a\u4e09\u89d2\u51fd\u6570\u7684\u534a\u89d2\u516c\u5f0f\uff08\u9644\u8bc1\u660e\uff09\u3002<\/p>\n\n\n\n \u2014\u2014\u6b63\/\u4f59\u5f26\u534a\u89d2\u516c\u5f0f<\/p>\n\n\n\n \u6839\u636e\u4f59\u5f26\u500d\u89d2\u516c\u5f0f\u7684\u53d8\u5f62\u5f0f $\\cos2\\theta=2\\cos^2\\theta-1$\uff0c\u5c06 ${2}\\theta$ \u6362\u6210 $\\theta$\uff0c$\\theta$ \u6362\u6210 $\\frac{\\theta}{2}$ \u5373\u5f97 $\\cos\\theta=2\\cos^2\\frac{\\theta}{2}-1$\uff0c$\\cos\\frac{\\theta}{2}=\\sqrt{\\frac{\\cos\\theta+1}{2}}$\u3002<\/p>\n\n\n\n \u540c\u7406\uff0c\u5bf9\u4e8e\u6b63\u5f26\u51fd\u6570\uff0c\u6709 $\\cos2\\theta=1-2\\sin^2\\theta\\rightarrow\\cos\\theta=1-2\\sin^2\\frac{\\theta}{2}\\rightarrow\\sin\\frac{\\theta}{2}=\\sqrt{\\frac{1-\\cos\\theta}{2}}$\u3002<\/p>\n\n\n\n \u2014\u2014\u6b63\u5207\u534a\u89d2\u516c\u5f0f<\/p>\n\n\n\n \u7531\u6b63\u5207\u51fd\u6570\u5b9a\u4e49\u53ef\u5f97 $\\tan\\frac{\\theta}{2}=\\dfrac{\\sin\\frac{\\theta}{2}}{\\cos\\frac{\\theta}{2}}$\uff0c\u5229\u7528\u4e09\u89d2\u51fd\u6570\u7684\u5347\u5e42\uff0c\u4e5f\u5c31\u662f\u4e0a\u9762\u5bfc\u51fa\u6b63\u4f59\u5f26\u534a\u89d2\u516c\u5f0f\u65f6\u4f7f\u7528\u7684\u4f59\u5f26\u500d\u89d2\u516c\u5f0f\uff0c\u5206\u5b50\u5206\u6bcd\u540c\u4e58 $\\cos\\frac{\\theta}{2}$ \u53ef\u5f97\uff1a$\\tan\\frac{\\theta}{2}=\\dfrac{\\sin\\frac{\\theta}{2}\\cos\\frac{\\theta}{2}}{\\cos^2\\frac{\\theta}{2}}=\\dfrac{\\frac{1}{2}\\sin\\theta}{\\frac{1}{2}(1+\\cos\\theta)}=\\dfrac{\\sin\\theta}{\\cos\\theta+1}$\u3002<\/p>\n\n\n\n \u56e0\u6b64\u8bc1\u660e\u9762\u79ef\u516c\u5f0f\u65f6\u51fa\u73b0\u7684 $\\frac{\\sin\\theta}{\\cos\\theta+1}$ \u53ef\u4ee5\u6362\u6210 $\\tan\\frac{\\theta}{2}$\u3002\u4e09\u4e2a\u516c\u5f0f\u6c47\u603b\u8d77\u6765\u5c31\u662f\uff1a<\/p>\n\n\n\n $$<\/p>\n\n\n\n \\sin\\frac{\\theta}{2}=\\sqrt{\\frac{1-\\cos\\theta}{2}}\\qquad\\cos\\frac{\\theta}{2}=\\sqrt{\\frac{\\cos\\theta+1}{2}}\\qquad\\tan\\frac{\\theta}{2}=\\frac{\\sin\\theta}{\\cos\\theta+1}<\/p>\n\n\n\n $$<\/p>\n\n\n\n #### \u5185\u5207\u5706<\/strong><\/p>\n\n\n\n > \u7126\u70b9\u4e09\u89d2\u5f62 $\\triangle PF_1F_2$ \u7684\u5185\u5207\u5706\u534a\u5f84\u4e3a $\\frac{c}{\\sin\\theta}$\uff0c\u5df2\u77e5\u534a\u5f84\u4e5f\u53ef\u6c42\u51fa\u9876\u89d2 $\\sin\\theta=\\frac{c}{R}$\u3002<\/p>\n\n\n\n **\u8bc4\u6790**<\/strong>\uff1a\u8fd9\u4e00\u6761\u5176\u5b9e\u4e5f\u6ca1\u4ec0\u4e48\uff0c\u4e3b\u8981\u662f\u6b63\u5f26\u5b9a\u7406\u7684\u8fd0\u7528\u3002\u56e0\u4e3a\u5728\u4e09\u89d2\u5f62\u4e2d $\\frac{a}{\\sin A}=\\frac{b}{\\sin B}=\\frac{c}{\\sin C}=2R$\uff0c\u5176\u4e2d $R$ \u5c31\u662f\u5185\u5207\u5706\u534a\u5f84\u3002\u5c06 $\\theta$ \u6240\u5bf9\u7684\u8fb9 $F_1F_2$ \u7684\u957f\u5ea6\u4ee3\u5165\u5373\u53ef\u8bc1\u5f97\u8be5\u7ed3\u8bba\u3002<\/p>\n\n\n\n #### \u79bb\u5fc3\u7387\u516c\u5f0f<\/strong><\/p>\n\n\n\n > \u4ee4\u7126\u70b9\u4e09\u89d2\u5f62 $\\triangle PF_1F_2$ \u7684\u5e95\u89d2 $\\angle PF_1F_2=\\alpha,\\angle PF_2F_1=\\beta$\uff0c\u90a3\u4e48\u692d\u5706\u7684\u79bb\u5fc3\u7387 $e=\\frac{\\sin(\\alpha+\\beta)}{sin\\alpha+\\sin\\beta}=\\frac{\\sin\\theta}{\\sin\\alpha+\\sin\\beta}$\u3002<\/p>\n\n\n\n **\u8bc4\u6790**<\/strong>\uff1a\u6709\u89d2\u6709\u8fb9\uff0c\u5f53\u7136\u8003\u8651\u6b63\u4f59\u5f26\u5b9a\u7406\u3002<\/p>\n\n\n\n **\u8bc1\u660e**<\/strong>\uff1a<\/p>\n\n\n\n \u6613\u77e5\u6b64\u65f6 $\\theta=\\pi-\\alpha-\\beta$\u3002\u6839\u636e\u6b63\u5f26\u5b9a\u7406\u6709 $\\frac{2c}{\\sin(\\pi-\\alpha-\\beta)}=\\frac{2c}{\\sin(\\alpha+\\beta)}=\\frac{|PF_1|}{\\sin\\beta}=\\frac{|PF_2|}{\\sin\\alpha}$\u3002\u5728\u692d\u5706\u4e2d\u53c8\u6709 $|PF_1|+|PF_2|=2a$\uff0c\u90a3\u4e48 $|PF_2|=2a-|PF_1|$\u3002<\/p>\n\n\n\n \u4ee3\u5165\u8fde\u7b49\u5f0f\u4e2d\uff0c\u5f97\u5230 $\\frac{2c}{\\sin(\\alpha+\\beta)}=\\frac{|PF_1|}{\\sin\\beta}=\\frac{2a-|PF_1|}{\\sin\\alpha}$\uff0c\u6839\u636e\u540e\u4e24\u9879\u53ef\u4ee5\u89e3\u51fa $|PF_1|=\\frac{2a\\sin\\beta}{\\sin\\alpha+\\sin\\beta}$\uff0c\u6b64\u65f6\u4ee3\u5165\u7b2c\u4e8c\u9879\u5f97\u5230 $\\frac{2c}{\\sin(\\alpha+\\beta)}=\\frac{2a}{\\sin\\alpha+\\sin\\beta}$\u3002<\/p>\n\n\n\n \u6b64\u65f6 $\\frac{c}{a}=e=\\frac{\\sin(\\alpha+\\beta)}{\\sin\\alpha+\\sin\\beta}=\\frac{\\sin\\theta}{\\sin\\alpha+\\sin\\beta}$\u3002<\/p>\n\n\n\n \u8bc1\u6bd5\u3002<\/p>\n\n\n\n ## \u53cc\u66f2\u7ebf \u57fa\u7840\u4e8c\u7ea7\u7ed3\u8bba<\/strong><\/p>\n\n\n\n \u82e5\u65e0\u7279\u6b8a\u8bf4\u660e\uff0c\u53cc\u66f2\u7ebf\u6807\u51c6\u65b9\u7a0b $E: \\frac{x^2}{a^2}-\\frac{y^2}{b^2}=1$\uff0c\u6ee1\u8db3 $a>0,b>0,a\\neq b$\uff0c\u7126\u70b9\u5728 $x$ \u8f74\u4e0a\u3002\u4e14\u82e5\u65e0\u7279\u6b8a\u8bf4\u660e\uff0c\u201c\u53cc\u66f2\u7ebf $E$\u201d\u5747\u6307\u4e0a\u8ff0\u7684\u6807\u51c6\u53cc\u66f2\u7ebf $E:\\frac{x^2}{a^2}-\\frac{y^2}{b^2}=1$\u3002<\/p>\n\n\n\n ### \u901a\u5f84<\/strong><\/p>\n\n\n\n > \u5728\u53cc\u66f2\u7ebf $E$ \u4e2d\uff0c\u4e0e\u7126\u70b9\u6240\u5728\u8f74\u5782\u76f4\u7684\u7126\u70b9\u5f26\u88ab\u53cc\u66f2\u7ebf\u622a\u5f97\u7ebf\u6bb5\u7684\u957f\u79f0\u4f5c\u5176\u901a\u5f84\u3002\u53cc\u66f2\u7ebf\u7684\u901a\u5f84\u957f $d=\\frac{2b^2}{a}$\u3002<\/p>\n\n\n\n **\u8bc4\u6790**<\/strong>\uff1a\u4e0e\u692d\u5706\u8bc1\u6cd5\u76f8\u540c\u3002<\/p>\n\n\n\n **\u8bc1\u660e**<\/strong>\uff1a<\/p>\n\n\n\n \u5c06\u6a2a\u5750\u6807 $\\pm c$ \u4ee3\u5165\u5f97 $y^2=b^2e^2-b^2$\u3002\u56e0\u4e3a\u53cc\u66f2\u7ebf\u6ee1\u8db3 $c^2=a^2+b^2$\uff0c\u53ef\u4ee5\u63a8\u5bfc\u51fa $e=\\sqrt{1+\\frac{b^2}{a^2}}$\u3002\u90a3\u4e48 $y^2=\\frac{b^4}{a^2}$\uff0c\u5f97\u5230 $y=\\pm\\frac{2b^2}{a}$\u3002\u6b64\u65f6\u901a\u5f84\u957f\u4e3a $d=2|y|=\\frac{2b^2}{a}$\u3002<\/p>\n\n\n\n \u8bc1\u6bd5\u3002<\/p>\n\n\n\n ### \u5706\u5468\u5b9a\u7406<\/strong><\/p>\n\n\n\n > \u5728\u53cc\u66f2\u7ebf $E$ \u4e2d\uff0c$A,B$ \u662f\u53cc\u66f2\u7ebf\u4e0a\u5173\u4e8e\u539f\u70b9\u5bf9\u79f0\u7684\u4e24\u70b9\uff0c$M$ \u662f\u53cc\u66f2\u7ebf\u4e0a\u5f02\u4e8e $A,B$ \u7684\u4e00\u70b9\u3002\u90a3\u4e48\u76f4\u7ebf $AM,BM$ \u7684\u659c\u7387\u4e4b\u79ef\u4e3a $\\frac{b^2}{a^2}$\u3002<\/p>\n\n\n\n **\u8bc4\u6790**<\/strong>\uff1a\u53cc\u66f2\u7ebf\u6709\u5173\u4e8c\u7ea7\u7ed3\u8bba\u7684\u8bc1\u660e\u601d\u8def\u548c\u692d\u5706\u57fa\u672c\u76f8\u540c\uff0c\u8fd9\u91cc\u6211\u4eec\u6cbf\u7528\u692d\u5706\u7684\u8bc1\u660e\u65b9\u6cd5\u7ee7\u7eed\u786c\u7b97\u3002<\/p>\n\n\n\n **\u8bc1\u660e**<\/strong>\uff1a<\/p>\n\n\n\n \u8bbe $A(x_1,y_1)~B(-x_1,-y_1)~M(x_2,y_2)$\uff0c\u90a3\u4e48\uff1a<\/p>\n\n\n\n $$<\/p>\n\n\n\n \\begin{aligned}<\/p>\n\n\n\n k_{AM}\\cdot k_{BM}&=\\frac{y_2-y_1}{x_2-x_1}\\times\\frac{y_2+y_1}{x_2+x_1}<\/p>\n\n\n\n \\\\&=\\frac{y_2^2-y_1^2}{x_2^2-x_1^2}<\/p>\n\n\n\n \\end{aligned}<\/p>\n\n\n\n $$<\/p>\n\n\n\n \u4e09\u70b9\u90fd\u5728\u53cc\u66f2\u7ebf\u4e0a\uff0c\u5f97\u5230\uff1a<\/p>\n\n\n\n $$<\/p>\n\n\n\n \\begin{cases}<\/p>\n\n\n\n y_1^2=\\dfrac{b^2x_1^2}{a^2}-b^2<\/p>\n\n\n\n \\\\\\qquad<\/p>\n\n\n\n \\\\y_2^2=\\dfrac{b^2x_2^2}{a^2}-b^2<\/p>\n\n\n\n \\end{cases}<\/p>\n\n\n\n $$<\/p>\n\n\n\n \u4ee3\u5165\u5f97\uff1a<\/p>\n\n\n\n $$<\/p>\n\n\n\n \\begin{aligned}<\/p>\n\n\n\n &=\\dfrac{\\frac{b^2}{a^2}(x_2^2-x_1^2)}{x_2^2-x_1^2}<\/p>\n\n\n\n \\\\&=\\frac{b^2}{a^2}<\/p>\n\n\n\n \\end{aligned}<\/p>\n\n\n\n $$<\/p>\n\n\n\n \u8bc1\u6bd5\u3002<\/p>\n\n\n\n **\u62d3\u5c55\u53d8\u5f62**<\/strong>\uff1a\u7126\u70b9\u6240\u5728\u5750\u6807\u8f74\u6539\u53d8\u65f6\u540c\u6837\u8981\u53d8\u6210 $\\frac{a^2}{b^2}$\u3002<\/p>\n\n\n\n ### \u5e7f\u4e49\u5782\u5f84\u5b9a\u7406\/\u4e2d\u70b9\u5f26\u516c\u5f0f<\/strong><\/p>\n\n\n\n > \u5728\u53cc\u66f2\u7ebf $E$ \u4e2d\uff0c$A,B$ \u4e3a\u53cc\u66f2\u7ebf\u4e0a\u4e24\u70b9\uff0c$M$ \u4e3a\u5f26 $AB$ \u7684\u4e2d\u70b9\uff0c\u90a3\u4e48\u76f4\u7ebf $OM$ \u4e0e\u76f4\u7ebf $AB$ \u7684\u659c\u7387\u4e4b\u79ef\u4e3a $\\frac{b^2}{a^2}$\u3002<\/p>\n\n\n\n **\u8bc4\u6790**<\/strong>\uff1a\u540c\u6837\u4f7f\u7528\u692d\u5706\u7684\u8bc1\u660e\u65b9\u6cd5<\/p>\n\n\n\n **\u8bc1\u660e**<\/strong>\uff1a<\/p>\n\n\n\n \u4ee4\u76f4\u7ebf $AB: y=kx+m$\uff0c\u659c\u7387\u4e0d\u5b58\u5728\u65f6\u65e0\u610f\u4e49\uff0c\u6545\u659c\u7387\u5b58\u5728\u3002\u8054\u7acb\u76f4\u7ebf\u548c\u53cc\u66f2\u7ebf\u65b9\u7a0b\uff1a<\/p>\n\n\n\n $$<\/p>\n\n\n\n \\begin{cases}<\/p>\n\n\n\n y=kx+m<\/p>\n\n\n\n \\\\\\frac{x^2}{a^2}-\\frac{y^2}{b^2}=1<\/p>\n\n\n\n \\end{cases}<\/p>\n\n\n\n $$<\/p>\n\n\n\n \u5f97\u5230\uff1a$(b^2-a^2k^2)x^2-2a^2mkx-a^2m^2-a^2b^2=0$\u3002\u97e6\u8fbe\u5b9a\u7406\u5f97 $x_1+x_2=\\dfrac{2a^2mk}{b^2-a^2k^2},x_1x_2=-\\dfrac{a^2m^2+a^2b^2}{b^2-a^2k^2}$\u3002<\/p>\n\n\n\n \u5f97\u5230\u4e2d\u70b9\u5750\u6807 $M\\left(\\dfrac{a^2mk}{b^2-a^2k^2},\\dfrac{b^2m}{b^2-a^2k^2}\\right)$\uff0c\u6b64\u65f6\u659c\u7387\u4e4b\u79ef\u8868\u793a\u4e3a\uff1a<\/p>\n\n\n\n $$<\/p>\n\n\n\n \\begin{aligned}<\/p>\n\n\n\n k_{AB}\\cdot k_{OM}&=k\\cdot\\dfrac{b^2m}{a^2mk}<\/p>\n\n\n\n \\\\&=k\\cdot\\dfrac{b^2}{a^2k}<\/p>\n\n\n\n \\\\&=\\frac{b^2}{a^2}<\/p>\n\n\n\n \\end{aligned}<\/p>\n\n\n\n $$<\/p>\n\n\n\n \u8bc1\u6bd5\u3002<\/p>\n\n\n\n **\u62d3\u5c55\u53d8\u5f62**<\/strong>\uff1a\u7126\u70b9\u5728 $y$ \u8f74\u4e0a\u65f6\u5bf9\u5e94\u7684\u4e58\u79ef\u662f $\\frac{a^2}{b^2}$\u3002<\/p>\n\n\n\n ### \u7126\u534a\u5f84\u516c\u5f0f<\/strong><\/p>\n\n\n\n > \u4ee4 $F_1,F_2$ \u4e3a\u53cc\u66f2\u7ebf $E$ \u7684\u5de6\u53f3\u7126\u70b9\uff0c$P(x_0,y_0)$ \u4e3a\u53cc\u66f2\u7ebf\u4e0a\u4e00\u70b9\uff0c$P$ \u5728\u53f3\u652f\u4e0a\u65f6\u6709 $|PF_1|=a+ex_0,|PF_2|=-a+ex_0$\uff1b\u5728\u5de6\u652f\u4e0a\u65f6\u6709 $|PF_1|=-a-ex_0,|PF_2|=a-ex_0$\u3002<\/p>\n\n\n\n **\u8bc4\u6790**<\/strong>\uff1a\u53cc\u66f2\u7ebf\u7b2c\u4e8c\u5b9a\u4e49\u7684\u53d8\u5f62<\/p>\n\n\n\n **\u8bc1\u660e**<\/strong>\uff1a<\/p>\n\n\n\n \u5f53 $P$ \u5728\u53f3\u652f\u65f6\uff0c\u6839\u636e\u7b2c\u4e8c\u5b9a\u4e49\uff0c\u6709 $\\dfrac{|PF_1|}{\\frac{a^2}{c}+x_0}=\\dfrac{|PF_1|}{\\frac{a}{e}+x_0}=e$\uff0c\u5f97\u5230 $|PF_1|=a+ex_0$\uff0c\u7136\u540e\u6839\u636e\u53cc\u66f2\u7ebf\u4e2d $||PF_1|-|PF_2||=2a$ \u53ef\u5f97 $|PF_2|=-a+ex_0$\u3002<\/p>\n\n\n\n \u540c\u7406\u53ef\u4ee5\u8bc1\u5f97\u5de6\u652f\u516c\u5f0f\u3002<\/p>\n\n\n\n \u8bc1\u6bd5\u3002<\/p>\n\n\n\n **\u62d3\u5c55\u53d8\u5f62**<\/strong>\uff1a\u7126\u70b9\u5728 $y$ \u8f74\u4e0a\u65f6\u8981\u628a $x_0$ \u6362\u6210 $y_0$\u3002<\/p>\n\n\n\n ### \u6e10\u8fd1\u7ebf\u76f8\u5173<\/strong><\/p>\n\n\n\n > \u8fc7\u539f\u70b9\u4e14\u5728\u65e0\u7a77\u8fdc\u5904\u4e0e\u53cc\u66f2\u7ebf\u7684\u8ddd\u79bb\u65e0\u9650\u8d8b\u8fd1\u4e8e ${0}$ \u7684\u4e24\u6761\u76f4\u7ebf\u53eb\u505a\u8fd9\u4e2a\u53cc\u66f2\u7ebf\u7684\u6e10\u8fd1\u7ebf\u3002\u7126\u70b9\u5728 $x$ \u8f74\u4e0a\u65f6\u6e10\u8fd1\u7ebf\u7684\u89e3\u6790\u5f0f\u4e3a $y=\\pm\\frac{b}{a}x$\uff1b\u82e5\u5728 $y$ \u8f74\u4e0a\u5219\u4e3a $y=\\pm\\frac{a}{b}x$\uff0c\u5373 $x=\\pm\\frac{b}{a}y$\u3002<\/p>\n\n\n\n #### \u7126\u70b9-\u6e10\u8fd1\u7ebf\u8ddd\u79bb<\/strong><\/p>\n\n\n\n > \u53cc\u66f2\u7ebf\u7684\u7126\u70b9\u4e0e\u4efb\u610f\u4e00\u6761\u6e10\u8fd1\u7ebf\u7684\u8ddd\u79bb\u5747\u4e3a $b$\u3002<\/p>\n\n\n\n **\u8bc4\u6790**<\/strong>\uff1a\u4f7f\u7528\u70b9\u5230\u76f4\u7ebf\u7684\u8ddd\u79bb\u516c\u5f0f\u8bc1\u660e\u3002<\/p>\n\n\n\n **\u8bc1\u660e**<\/strong>\uff1a<\/p>\n\n\n\n \u5de6\u7126\u70b9 $F_1(-c,0)$\uff0c\u5230\u6e10\u8fd1\u7ebf $y\\pm\\frac{b}{a}x=0$ \u7684\u8ddd\u79bb\u4e3a\uff1a<\/p>\n\n\n\n $$<\/p>\n\n\n\n d=\\dfrac{\\frac{b}{a}c}{\\sqrt{1+\\frac{b^2}{a^2}}}=\\dfrac{eb}{e}=b<\/p>\n\n\n\n $$<\/p>\n\n\n\n \u8bc1\u6bd5\u3002<\/p>\n\n\n\n ### \u7126\u70b9\u4e09\u89d2\u5f62\u76f8\u5173<\/strong><\/p>\n\n\n\n > \u53cc\u66f2\u7ebf\u7684\u5de6\u53f3\u7126\u70b9 $F_1,F_2$ \u4e0e\u53cc\u66f2\u7ebf\u4e0a\u4e00\u70b9 $P$ \u7ec4\u6210\u7684\u4e09\u89d2\u5f62 $\\triangle PF_1F_2$ \u79f0\u4f5c\u8fd9\u4e2a\u53cc\u66f2\u7ebf\u7684\u7126\u70b9\u4e09\u89d2\u5f62\u3002<\/p>\n\n\n\n \u672c\u8282\u4e2d\u51fa\u73b0\u7684\u89d2 $\\theta$ \u82e5\u65e0\u7279\u6b8a\u8bf4\u660e\u5747\u6307\u4ee3 $\\angle F_1PF_2$\u3002<\/p>\n\n\n\n #### \u5468\u957f<\/strong><\/p>\n\n\n\n > \u7126\u70b9\u4e09\u89d2\u5f62 $\\triangle PF_1F_2$ \u7684\u5468\u957f\u4e3a ${2}e|x_0|+2c$\u3002<\/p>\n\n\n\n **\u8bc4\u6790**<\/strong>\uff1a\u6839\u636e\u524d\u9762\u6240\u8bc1\u660e\u7684\u7126\u534a\u5f84\u516c\u5f0f\u53ef\u5f97\u8fd9\u4e2a\u7ed3\u8bba\u3002<\/p>\n\n\n\n #### \u9762\u79ef<\/strong><\/p>\n\n\n\n > \u7126\u70b9\u4e09\u89d2\u5f62 $\\triangle PF_1F_2$ \u7684\u9762\u79ef\u4e3a $\\dfrac{b^2}{\\tan\\frac{\\theta}{2}}=b^2\\cot\\frac{\\theta}{2}$\u3002<\/p>\n\n\n\n **\u8bc1\u660e**<\/strong>\uff1a<\/p>\n\n\n\n \u6b63\u5f26\u5b9a\u7406\u5f97\uff1a$S=\\frac{1}{2}|PF_1||PF_2|\\sin\\theta$\u3002\u4f59\u5f26\u5b9a\u7406\u5f97\uff1a${4}c^2=|PF_1|^2+|PF_2|^2-2|PF_1||PF_2|\\cos\\theta$\uff0c\u5f97\u5230 $|PF_1||PF_2|=\\dfrac{|PF_1|^2+|PF_2|^2-4c^2}{2\\cos\\theta}$\u3002\u6839\u636e\u5b8c\u5168\u5e73\u65b9\u516c\u5f0f\uff0c\u6709 $(|PF_1|-|PF_2|)^2=4a^2=|PF_1|^2+|PF_2|^2-2|PF_1||PF_2|$\uff0c\u8054\u7acb\u53ef\u5f97 $|PF_1||PF_2|=\\dfrac{|PF_1||PF_2|+2a^2-2c^2}{\\cos\\theta}$\uff0c\u89e3\u5f97 $|PF_1||PF_2|=\\dfrac{2b^2}{1-\\cos\\theta}$\u3002\u4ee3\u5165\u9762\u79ef\u516c\u5f0f\u5f97 $S=\\dfrac{b^2\\sin\\theta}{1-\\cos\\theta}=b^2\\cot\\frac{\\theta}{2}$\u3002<\/p>\n\n\n\n \u8bc1\u6bd5\u3002<\/p>\n\n\n\n **\u62d3\u5c55\u53d8\u5f62**<\/strong>\uff1a\u4f59\u5207\u7684\u534a\u89d2\u516c\u5f0f\u8bc1\u660e\u3002<\/p>\n\n\n\n $$<\/p>\n\n\n\n \\begin{aligned}<\/p>\n\n\n\n \\cot\\frac{\\theta}{2}&=\\dfrac{\\cos\\frac{\\theta}{2}}{\\sin\\frac{\\theta}{2}}<\/p>\n\n\n\n \\\\&=\\dfrac{\\sin\\frac{\\theta}{2}\\cos\\frac{\\theta}{2}}{\\sin^2\\frac{\\theta}{2}}<\/p>\n\n\n\n \\\\&=\\dfrac{\\frac{1}{2}\\sin\\theta}{\\frac{1}{2}(1-\\cos\\theta)}<\/p>\n\n\n\n \\\\&=\\dfrac{\\sin\\theta}{1-\\cos\\theta}<\/p>\n\n\n\n \\end{aligned}<\/p>\n\n\n\n $$<\/p>\n\n\n\n #### \u79bb\u5fc3\u7387\u516c\u5f0f<\/strong><\/p>\n\n\n\n > \u4ee4\u7126\u70b9\u4e09\u89d2\u5f62 $\\triangle PF_1F_2$ \u7684\u5e95\u89d2 $\\angle PF_1F_2=\\alpha,\\angle PF_2F_1=\\beta$\uff0c\u90a3\u4e48\u53cc\u66f2\u7ebf\u7684\u79bb\u5fc3\u7387\u4e3a $e=\\dfrac{\\sin\\theta}{\\sin\\beta-\\sin\\alpha}$\u3002<\/p>\n\n\n\n **\u8bc1\u660e**<\/strong>\uff1a<\/p>\n\n\n\n \u7531\u6b63\u5f26\u5b9a\u7406\uff0c$\\frac{2c}{\\sin\\theta}=\\frac{|PF_1|}{\\sin\\beta}=\\frac{|PF_2|}{\\sin\\alpha}$\uff0c\u4e0d\u59a8\u5047\u8bbe\u5f53\u524d $P$ \u5728\u53f3\u652f\u4e0a\uff0c\u90a3\u4e48 $|PF_1|-|PF_2|=2a$\uff0c\u5373 $|PF_2|=|PF_1|-2a$\u3002\u89e3\u5f97 $|PF_1|=\\frac{2a\\sin\\beta}{\\sin\\beta-\\sin\\alpha}$\u3002\u6b64\u65f6\u6709 $\\frac{2c}{\\sin\\theta}=\\frac{2a}{\\sin\\beta-\\sin\\alpha}$\u3002\u5f97\u5230 $e=\\frac{c}{a}=\\frac{\\sin\\theta}{\\sin\\beta-\\sin\\alpha}$\u3002<\/p>\n\n\n\n \u8bc1\u6bd5\u3002<\/p>\n\n\n\n ## \u629b\u7269\u7ebf\u57fa\u7840\u4e8c\u7ea7\u7ed3\u8bba<\/strong><\/p>\n\n\n\n \u82e5\u65e0\u7279\u6b8a\u8bf4\u660e\uff0c\u629b\u7269\u7ebf\u6807\u51c6\u65b9\u7a0b $E: y^2=2px$ \u5747\u6ee1\u8db3 $p>0$\uff0c\u7126\u70b9\u5728 $x$ \u8f74\u6b63\u534a\u8f74\u3002\u4e14\u82e5\u65e0\u7279\u6b8a\u6307\u660e\uff0c\u201c\u629b\u7269\u7ebf $E$\u201d\u5747\u6307\u4e0a\u8ff0\u7684\u629b\u7269\u7ebf $E:y^2=2px$\u3002<\/p>\n\n\n\n ### \u901a\u5f84<\/strong><\/p>\n\n\n\n > \u629b\u7269\u7ebf\u7684\u901a\u5f84\u957f\u4e3a $2p$\u3002<\/p>\n\n\n\n **\u8bc4\u6790**<\/strong>\uff1a\u629b\u7269\u7ebf\u4e2d\u53ea\u8981\u6d89\u53ca\u5230\u7126\u534a\u5f84\u76f8\u5173\u7684\u5185\u5bb9\uff0c\u90fd\u8981\u7b2c\u4e00\u65f6\u95f4\u60f3\u5230\u7126\u534a\u5f84\u957f\u7b49\u4e8e\u8be5\u70b9\u4e0e\u51c6\u7ebf\u7684\u8ddd\u79bb\u4ece\u800c\u8fdb\u884c\u8f6c\u5316\uff0c\u8fd9\u6837\u53ef\u4ee5\u7b80\u5316\u8ba1\u7b97\u3002<\/p>\n\n\n\n **\u8bc1\u660e**<\/strong>\uff1a<\/p>\n\n\n\n \u6a2a\u5750\u6807 $\\frac{p}{2}$ \u4ee3\u5165\uff0c\u5f97\u5230\u7126\u534a\u5f84\u4e3a $\\frac{p}{2}+\\frac{p}{2}=p$\uff0c\u901a\u5f84\u4e3a\u4e8c\u500d\u7126\u534a\u5f84\uff0c\u5373 $2p$\u3002<\/p>\n\n\n\n \u8bc1\u6bd5\u3002<\/p>\n\n\n\n ### \u7126\u70b9\u5f26\u5b9a\u7406<\/strong><\/p>\n\n\n\n > \u629b\u7269\u7ebf $E$ \u7684\u4e00\u6761\u7126\u70b9\u5f26\u4ea4\u629b\u7269\u7ebf\u4e8e $A,B$ \u4e24\u70b9\uff0c\u90a3\u4e48\u76f4\u7ebf $OA$ \u4e0e\u76f4\u7ebf $OB$ \u7684\u4e58\u79ef\u4e3a\u5b9a\u503c $-4$\u3002<\/p>\n\n\n\n **\u8bc4\u6790**<\/strong>\uff1a\u6211\u4eec\u53ef\u4ee5\u6070\u5f53\u9009\u62e9\u76f4\u7ebf\u7684\u6a2a\u622a\u5f0f\u548c\u659c\u622a\u5f0f\u6765\u65b9\u4fbf\u8ba1\u7b97\u3002\u5728\u672c\u4f8b\u4e2d\uff0c\u7531\u4e8e\u629b\u7269\u7ebf\u65b9\u7a0b\u7684\u4e8c\u6b21\u9879\u5728 $y$ \u4e0a\uff0c\u5e76\u4e14\u76f4\u7ebf\u8fc7 $x$ \u8f74\u4e0a\u7684\u5b9a\u70b9\uff0c\u6211\u4eec\u81ea\u7136\u5730\u9009\u62e9\u6a2a\u622a\u5f0f\u6765\u8fdb\u884c\u8ba1\u7b97\u3002<\/p>\n\n\n\n **\u8bc1\u660e**<\/strong>\uff1a<\/p>\n\n\n\n \u4ee4\u76f4\u7ebf $AB: x=ty+\\frac{p}{2}$\uff0c\u8054\u7acb\u629b\u7269\u7ebf\u65b9\u7a0b $y^2=2px$ \u5f97\uff1a<\/p>\n\n\n\n $$<\/p>\n\n\n\n \\begin{aligned}<\/p>\n\n\n\n y^2&=2pty+p^2<\/p>\n\n\n\n \\\\y^2-2pty-p^2&=0<\/p>\n\n\n\n \\end{aligned}<\/p>\n\n\n\n $$<\/p>\n\n\n\n \u6839\u636e\u97e6\u8fbe\u5b9a\u7406\uff0c\u5f97\uff1a<\/p>\n\n\n\n $$<\/p>\n\n\n\n y_1+y_2=2pt\\qquad y_1y_2=-p^2<\/p>\n\n\n\n \\\\x_1+x_2=2t(y_1+y_2)+p=4t^2p+p\\qquad x_1x_2=t^2y_1y_2+\\frac{pt}{2}(y_1+y_2)+\\frac{p^2}{4}=\\frac{p^2}{4}<\/p>\n\n\n\n $$<\/p>\n\n\n\n \u659c\u7387\u7684\u4e58\u79ef\u8868\u793a\u4e3a\uff1a<\/p>\n\n\n\n $$<\/p>\n\n\n\n \\begin{aligned}<\/p>\n\n\n\n k_{OA}\\cdot k_{OB}&=\\frac{y_1y_2}{x_1x_2}<\/p>\n\n\n\n \\\\&=-\\dfrac{p^2}{\\frac{p^2}{4}}<\/p>\n\n\n\n \\\\&=-4<\/p>\n\n\n\n \\end{aligned}<\/p>\n\n\n\n $$<\/p>\n\n\n\n \u8bc1\u6bd5\u3002<\/p>\n\n\n\n **\u62d3\u5c55\u53d8\u5f62**<\/strong>\uff1a\u4e8b\u5b9e\u4e0a\uff0c\u8bc1\u660e\u8fc7\u7a0b\u4e2d\u7531\u97e6\u8fbe\u5b9a\u7406\u5bfc\u51fa\u7684\u5173\u7cfb\u5f0f $x_1x_2=\\frac{p^2}{4}$ \u548c $y_1y_2=-p^2$ \u5728\u5b9e\u8df5\u4e2d\u66f4\u4e3a\u5e38\u7528\u4e00\u4e9b\u3002<\/p>\n\n\n\n ### \u4e24\u70b9\u5f26\u516c\u5f0f<\/strong><\/p>\n\n\n\n > \u629b\u7269\u7ebf $E$ \u4e0a\u4e24\u70b9 $A(x_1,y_1)$ \u548c $B(x_2,y_2)$ \u7ec4\u6210\u7684\u5f26 $AB$ \u7684\u659c\u7387\u4e3a $\\frac{2p}{y_1+y_2}$\u3002<\/p>\n\n\n\n **\u8bc4\u6790**<\/strong>\uff1a\u9002\u65f6\u907f\u5f00\u7e41\u7410\u7684\u9ad8\u6b21\u8ba1\u7b97\u662f\u975e\u5e38\u6709\u7528\u7684\u3002<\/p>\n\n\n\n **\u8bc1\u660e**<\/strong>\uff1a<\/p>\n\n\n\n \u56e0\u4e3a\u4e24\u70b9\u5728\u629b\u7269\u7ebf\u4e0a\uff0c\u56e0\u6b64\u5750\u6807\u6ee1\u8db3\uff1a<\/p>\n\n\n\n $$<\/p>\n\n\n\n \\begin{cases}<\/p>\n\n\n\n y_1^2=2px_1<\/p>\n\n\n\n \\\\y_2^2=2px_2<\/p>\n\n\n\n \\end{cases}\\rightarrow\\begin{cases}<\/p>\n\n\n\n x_1=\\dfrac{y_1^2}{2p}<\/p>\n\n\n\n \\\\\\qquad<\/p>\n\n\n\n \\\\x_2=\\dfrac{y_2^2}{2p}<\/p>\n\n\n\n \\end{cases}<\/p>\n\n\n\n $$<\/p>\n\n\n\n \u6240\u4ee5\u659c\u7387\u53ef\u4ee5\u8868\u793a\u4e3a\uff1a$\\dfrac{y_2-y_1}{x_2-x_1}=\\dfrac{y_2-y_1}{\\frac{y_2^2-y_1^2}{2p}}=\\dfrac{2p(y_2-y_1)}{(y_1+y_2)(y_2-y_1)}=\\dfrac{2p}{y_1+y_2}$\u3002<\/p>\n\n\n\n \u8bc1\u6bd5\u3002<\/p>\n\n\n\n ### \u7126\u534a\u5f84\u516c\u5f0f<\/strong><\/p>\n\n\n\n > \u629b\u7269\u7ebf $E$ \u7684\u7126\u70b9\u5f26 $AB$ \u5206\u522b\u5728\u7b2c\u4e00\u8c61\u9650\u548c\u7b2c\u56db\u8c61\u9650\u4ea4\u629b\u7269\u7ebf\u4e8e $A,B$ \u4e24\u70b9\uff0c\u76f4\u7ebf $AB$ \u4e0e $x$ \u8f74\u7684\u5939\u89d2\u662f $\\theta$\uff0c\u90a3\u4e48 $|AF|=\\dfrac{p}{1-\\cos\\theta},|BF|=\\dfrac{p}{1+\\cos\\theta},|AB|=\\dfrac{2p}{\\sin^2\\theta}$\u3002<\/p>\n\n\n\n **\u8bc4\u6790**<\/strong>\uff1a\u76f4\u63a5\u770b\u4e0d\u592a\u5bb9\u6613\uff0c\u6765\u4e00\u5f20\u56fe\u8f85\u52a9\u4e00\u4e0b\uff1a<\/p>\n\n\n\n <\/p>\n\n\n\n **\u8bc1\u660e**<\/strong>\uff1a<\/p>\n\n\n\n \u7531\u629b\u7269\u7ebf\u5b9a\u4e49\u77e5\uff1a$|AH|=|AF|=|GH|+|AG|=|GH|+|AF|\\cos\\theta=p+|AF|\\cos\\theta$\uff0c\u79fb\u9879\u53ef\u5f97 $|AF|=\\dfrac{p}{1-\\cos\\theta}$\u3002\u540c\u7406\u53ef\u8bc1\u5f97 $|BF|=\\frac{p}{1+\\cos\\theta}$\u3002<\/p>\n\n\n\n \u6b64\u65f6 $|AB|=|AF|+|BF|=\\dfrac{p}{1-\\cos\\theta}+\\dfrac{p}{1+\\cos\\theta}=\\dfrac{2p}{\\sin^2\\theta}$\u3002<\/p>\n\n\n\n \u8bc1\u6bd5\u3002<\/p>\n\n\n\n ## \u692d\u5706-\u53cc\u66f2\u7ebf\u5171\u7126\u70b9\u95ee\u9898<\/strong><\/p>\n\n\n\n \u5728\u672c\u7ae0\u4e2d\uff0c\u6211\u4eec\u9ed8\u8ba4\u5b58\u5728\u4e00\u4e2a\u692d\u5706 $E_1: \\frac{x^2}{a_1^2}+\\frac{y^2}{b_1^2}=1$ \u4e0e $E_2:\\frac{x^2}{a_2^2}-\\frac{y^2}{b_2^2}=1$ \u5171\u7126\u70b9\u3002\u82e5\u65e0\u7279\u6b8a\u8bf4\u660e\uff0c$P$ \u4e3a\u4e24\u5706\u9525\u66f2\u7ebf\u5728\u7b2c\u4e00\u8c61\u9650\u5185\u7684\u4ea4\u70b9\uff0c$\\angle F_1PF_2=\\theta$\u3002\u5982\u4e0b\u56fe\uff1a<\/p>\n\n\n\n <\/p>\n\n\n\n ### \u7126\u534a\u5f84<\/strong><\/p>\n\n\n\n > \u5171\u7126\u70b9\u7684\u692d\u5706\u548c\u53cc\u66f2\u7ebf\u6ee1\u8db3 $|PF_1|=a_1+a_2,|PF_2|=a_1-a_2$\u3002<\/p>\n\n\n\n **\u8bc4\u6790**<\/strong>\uff1a\u6ce8\u610f\u5229\u7528\u597d\u692d\u5706\u548c\u53cc\u66f2\u7ebf\u7684\u5b9a\u4e49\u3002<\/p>\n\n\n\n **\u8bc1\u660e**<\/strong>\uff1a<\/p>\n\n\n\n \u5728\u692d\u5706\u4e2d\uff0c\u6709 $|PF_1|+|PF_2|=2a_1$\uff1b\u5728\u53cc\u66f2\u7ebf\u4e2d\uff0c\u6709 $|PF_1|-|PF_2|=2a_2$\uff0c\u4e24\u5f0f\u76f8\u52a0\u5f97 ${2}|PF_1|=2a_1+2a_2$\uff0c\u76f8\u51cf\u5f97 ${2}|PF_2|=2a_1-2a_2$\u3002\u7531\u6b64\u5f97\u5230\uff1a<\/p>\n\n\n\n $$<\/p>\n\n\n\n |PF_1|=a_1+a_2\\qquad |PF_2|=a_1-a_2<\/p>\n\n\n\n $$<\/p>\n\n\n\n \u8bc1\u6bd5\u3002<\/p>\n\n\n\n ### \u79bb\u5fc3\u7387\u4e0e\u89d2\u7684\u5173\u7cfb<\/strong><\/p>\n\n\n\n > \u5171\u7126\u70b9\u7684\u692d\u5706\u548c\u53cc\u66f2\u7ebf\u6ee1\u8db3 $\\dfrac{\\sin^2\\frac{\\theta}{2}}{e_1^2}+\\dfrac{\\cos^2\\frac{\\theta}{2}}{e_2^2}=1$<\/p>\n\n\n\n **\u8bc4\u6790**<\/strong>\uff1a\u540c\u6837\u662f\u6709\u89d2\u6709\u8fb9\uff0c\u8003\u8651\u6b63\/\u4f59\u5f26\u5b9a\u7406\u3002\u8fd9\u4e2a\u7ed3\u8bba\u53ef\u4ee5\u5e2e\u52a9\u4f60\u5feb\u901f\u89e3\u51b3\u8bf8\u5982 $e_1^2e_2^2,\\frac{1}{e_1^2}+\\frac{1}{e_2^2}$ \u7b49\u5f0f\u5b50\u7684\u6700\u503c\u95ee\u9898\u3002<\/p>\n\n\n\n **\u8bc1\u660e**<\/strong>\uff1a<\/p>\n\n\n\n \u501f\u7528\u4e0a\u4e00\u8282\u7684\u7126\u534a\u5f84\u7ed3\u8bba\uff0c\u5e76\u7efc\u5408\u4f59\u5f26\u5b9a\u7406\uff0c\u53ef\u4ee5\u5f97\u5230\uff1a<\/p>\n\n\n\n $$<\/p>\n\n\n\n \\begin{aligned}<\/p>\n\n\n\n a_1^2+a_2^2+2a_1a_2+a_1^2+a_2^2-2a_1a_2-2(a_1^2-a_2^2)\\cos\\theta&=4c^2<\/p>\n\n\n\n \\\\2a_1^2+2a_2^2-2a_1^2\\cos\\theta+2a_2^2\\cos\\theta&=4c^2<\/p>\n\n\n\n \\\\(1-\\cos\\theta)a_1^2+(1+\\cos\\theta)a_2^2&=2c^2<\/p>\n\n\n\n \\\\\\dfrac{(1-\\cos\\theta)a_1^2}{2c^2}+\\dfrac{(1+\\cos\\theta)a_2^2}{2c^2}&=1<\/p>\n\n\n\n \\\\\\dfrac{1-\\cos\\theta}{2e_1^2}+\\dfrac{1+\\cos\\theta}{2e_2^2}&=1<\/p>\n\n\n\n \\\\\\dfrac{\\sin^2\\frac{\\theta}{2}}{e_1^2}+\\dfrac{\\cos^2\\frac{\\theta}{2}}{e_2^2}&=1<\/p>\n\n\n\n \\end{aligned}<\/p>\n\n\n\n $$<\/p>\n\n\n\n \u8bc1\u6bd5\u3002<\/p>\n\n\n\n **\u62d3\u5c55\u53d8\u5f62**<\/strong>\uff1a\u6b63\u4f59\u5f26\u51fd\u6570\u7684\u5347\u5e42\/\u964d\u5e42\u516c\u5f0f\u3002<\/p>\n\n\n\n \u7531\u4f59\u5f26\u7684\u4e8c\u500d\u89d2\u516c\u5f0f $\\cos\\theta=2\\cos^2\\frac{\\theta}{2}-1=1-2\\sin^2\\frac{\\theta}{2}$\uff0c\u5f97\u5230 $\\sin^2\\frac{\\theta}{2}=\\frac{1-\\cos\\theta}{2},\\cos^2\\frac{\\theta}{2}=\\frac{1+\\cos\\theta}{2}$\u3002\u5373\u8bc1\u5f97\u964d\u5e42\u516c\u5f0f\u3002\u4e8b\u5b9e\u4e0a\uff0c\u4f59\u5f26\u7684\u4e8c\u500d\u89d2\u516c\u5f0f\u5c31\u662f\u5347\u5e42\u516c\u5f0f\u3002<\/p>\n\n\n\n ## \u8499\u65e5\u5706<\/strong><\/p>\n\n\n\n \u692d\u5706 $E$ \u4e0a\u4efb\u610f\u4e24\u6761\u4e92\u76f8\u5782\u76f4\u7684\u5207\u7ebf\u7126\u70b9\u7684\u8f68\u8ff9\u7ec4\u6210\u4e86\u4e00\u4e2a\u5706\uff0c\u79f0\u4f5c\u8499\u65e5\u5706\/\u5916\u51c6\u5706\u3002\u692d\u5706 $\\frac{x^2}{a^2}+\\frac{y^2}{b^2}=1$ \u7684\u8499\u65e5\u5706\u4e3a $x^2+y^2=a^2+b^2$\u3002\u5982\u4e0b\u56fe\uff1a<\/p>\n\n\n\n <\/p>\n\n\n\n \u82e5\u65e0\u7279\u6b8a\u8bf4\u660e\uff0c\u4e24\u5207\u7ebf\u4ea4\u4e8e $P$\uff0c\u4e14\u4e0e\u692d\u5706\u5206\u522b\u4ea4\u4e8e\u70b9 $A$ \u548c $B$\uff0c\u4e0e\u8499\u65e5\u5706\u5206\u522b\u4ea4\u4e8e\u70b9 $C$ \u548c $D$\u3002<\/p>\n\n\n\n ### \u8f68\u8ff9\u65b9\u7a0b<\/strong><\/p>\n\n\n\n > \u692d\u5706 $E$ \u5bf9\u5e94\u7684\u8499\u65e5\u5706\u65b9\u7a0b\u4e3a $x^2+y^2=a^2+b^2$\u3002<\/p>\n\n\n\n **\u8bc4\u6790**<\/strong>\uff1a\u6ca1\u6709\u611f\u60c5\uff0c\u53ea\u6709\u8bbe\u70b9\u3002<\/p>\n\n\n\n **\u8bc1\u660e**<\/strong>\uff1a<\/p>\n\n\n\n \u5207\u7ebf\u659c\u7387\u4e0d\u5b58\u5728\u65f6\uff0c$P(\\pm a,\\pm b)$\uff0c\u663e\u7136\u5728\u5706 $x^2+y^2=a^2+b^2$ \u4e0a\u3002<\/p>\n\n\n\n \u5207\u7ebf\u659c\u7387\u5b58\u5728\u65f6\uff0c\u8bbe $PM:y=kx+m$\uff0c\u5219\u6839\u636e\u5782\u76f4\u5173\u7cfb\u6709 $PN:y=-\\frac{1}{k}x+n$\u3002<\/p>\n\n\n\n \u8054\u7acb $PM$ \u4e0e\u692d\u5706\u65b9\u7a0b\uff0c\u5e76\u6839\u636e\u76f8\u5207\u5173\u7cfb\u5f97\uff1a<\/p>\n\n\n\n $$<\/p>\n\n\n\n \\begin{aligned}<\/p>\n\n\n\n (b^2+a^2k^2)x^2+2a^2mkx+a^2m^2-a^2b^2&=0<\/p>\n\n\n\n \\\\\\Delta&=0<\/p>\n\n\n\n \\\\4a^4m^2k^2-4a^2(b^2+a^2k^2)(m^2-b^2)&=0<\/p>\n\n\n\n \\\\a^2k^2+b^2-m^2&=0<\/p>\n\n\n\n \\\\m^2&=a^2k^2+b^2<\/p>\n\n\n\n \\end{aligned}<\/p>\n\n\n\n $$<\/p>\n\n\n\n \u540c\u7406\u53ef\u5f97 $\\frac{a^2}{k^2}+b^2-n^2=0\\rightarrow a^2+b^2k^2=n^2k^2$\u3002<\/p>\n\n\n\n \u8054\u7acb\u4e24\u76f4\u7ebf\u65b9\u7a0b\u5f97\u5230 $P\\left(\\frac{k(n-m)}{k^2+1},\\frac{nk^2+m}{k^2+1}\\right)$\uff0c\u6b64\u65f6 $|OP|^2=\\frac{n^2k^2+m^2}{k^2+1}=\\frac{a^2+b^2k^2+a^2k^2+b^2}{k^2+1}=a^2+b^2$\u3002\u56e0\u6b64 $P$ \u5728\u5706 $x^2+y^2=a^2+b^2$ \u4e0a\u3002<\/p>\n\n\n\n \u8bc1\u6bd5\u3002<\/p>\n\n\n\n ### \u51e0\u4f55\u6027\u8d28 \u5176\u4e00<\/strong><\/p>\n\n\n\n > \u8499\u65e5\u5706\u4e0a\u4e00\u70b9 $P$ \u5f15\u51fa\u7684\u4e24\u6761\u5207\u7ebf\u4ea4\u8499\u65e5\u5706\u4e8e $C,D$ \u4e24\u70b9\uff0c\u76f4\u7ebf $CD$ \u8fc7\u539f\u70b9\u3002<\/p>\n\n\n\n **\u8bc4\u6790**<\/strong>\uff1a\u65e0<\/p>\n\n\n\n **\u8bc1\u660e**<\/strong>\uff1a<\/p>\n\n\n\n \u6839\u636e\u5706\u5185\u76f4\u5f84\u6240\u5bf9\u7684\u5706\u5468\u89d2\u6052\u4e3a\u76f4\u89d2\u7684\u5173\u7cfb\uff0c\u53ef\u5f97 $CD$ \u4e3a\u8499\u65e5\u5706\u76f4\u5f84\uff0c\u5373 $C,O,D$ \u4e09\u70b9\u5171\u7ebf\u3001$CD$ \u8fc7\u539f\u70b9\u3002<\/p>\n\n\n\n \u8bc1\u6bd5\u3002<\/p>\n\n\n\n ### \u5e7f\u4e49\u5782\u5f84\u5b9a\u7406<\/strong><\/p>\n\n\n\n > $P$ \u4e3a\u8499\u65e5\u5706\u4e0a\u4e00\u70b9\uff0c\u8fc7 $P$ \u4f5c\u692d\u5706 $E$ \u7684\u4e24\u6761\u5207\u7ebf $PA,PB$\uff0c\u5207\u70b9\u4e3a $A,B$\uff0c\u8fde\u63a5 $OP$\uff0c\u5219 $k_{OP}\\cdot k_{AB}=-\\frac{b^2}{a^2}$\u3002<\/p>\n\n\n\n **\u8bc4\u6790**<\/strong>\uff1a\u5229\u7528\u5706\u9525\u66f2\u7ebf\u7684\u5207\u70b9\u5f26\u65b9\u7a0b\u5373\u53ef\u5feb\u901f\u89e3\u51b3\u3002<\/p>\n\n\n\n **\u8bc1\u660e**<\/strong>\uff1a<\/p>\n\n\n\n \u4ee4 $P(x_0,y_0)$\uff0c\u90a3\u4e48 $k_{OP}=\\frac{y_0}{x_0}$\u3002\u6839\u636e\u5706\u9525\u66f2\u7ebf\u7684\u5207\u70b9\u5f26\u516c\u5f0f\uff0c\u5f97\u5230\u5207\u70b9\u5f26 $AB:\\frac{x_0}{a^2}x+\\frac{y_0}{b^2}y=1$\uff0c\u5f97\u5230 $k_{AB}=-\\frac{b^2x_0}{a^2y_0}$\u3002\u76f8\u4e58\u5373\u5f97\u7ed3\u679c $-\\frac{b^2}{a^2}$\u3002<\/p>\n\n\n\n \u8bc1\u6bd5\u3002<\/p>\n\n\n\n **\u62d3\u5c55\u53d8\u5f62**<\/strong>\uff1a\u692d\u5706\u4ea4\u70b9\u6240\u5728\u5750\u6807\u8f74\u53d8\u5316\u540e\u4ecd\u7136\u4f1a\u53d8\u6210 $-\\frac{a^2}{b^2}$\u3002\u540c\u65f6\u6839\u636e\u7ed3\u679c\u548c\u4e2d\u70b9\u5f26\u516c\u5f0f\u53ef\u4ee5\u5f97\u77e5 $AB$ \u4e0e $OP$ \u7684\u4ea4\u70b9 $M$ \u4e3a $AB$ \u4e2d\u70b9\u3002<\/p>\n\n\n\n ### \u51e0\u4f55\u6027\u8d28 \u5176\u4e8c<\/strong><\/p>\n\n\n\n > \u8499\u65e5\u5706\u4e0a\u4e00\u70b9 $P$ \u5411\u692d\u5706\u5f15\u4e24\u6761\u5207\u7ebf $PA$ \u548c $PB$\uff0c\u4ea4\u692d\u5706\u4e8e $A,B$\uff0c\u4ea4\u8499\u65e5\u5706\u4e8e $C,D$\uff0c$OP$ \u4ea4 $AB$ \u4e8e $M$ \u70b9\uff0c\u6709 $AB\/\/CD$\u3002<\/p>\n\n\n\n **\u8bc4\u6790**<\/strong>\uff1a\u5229\u7528\u51e0\u4f55\u5173\u7cfb\u8fdb\u884c\u8bc1\u660e\u3002\u524d\u7f6e\u662f\u4e0a\u9762\u7684\u5e7f\u4e49\u5782\u5f84\u5b9a\u7406\u548c\u51e0\u4f55\u6027\u8d28\u4e00\u3002<\/p>\n\n\n\n **\u8bc1\u660e**<\/strong>\uff1a<\/p>\n\n\n\n \u6839\u636e\u8499\u65e5\u5706\uff0c\u5f97\u5230\u9876\u89d2 $\\angle APB=90\\degree$\u3002\u6839\u636e\u4e0a\u9762\u5e7f\u4e49\u5782\u5f84\u5b9a\u7406\u5f97\u5230\u7684 $M$ \u4e3a $AB$ \u4e2d\u70b9\u7684\u5173\u7cfb\uff0c\u7ed3\u5408\u76f4\u89d2\u4e09\u89d2\u5f62\u659c\u8fb9\u4e0a\u7684\u4e2d\u7ebf\u5b9a\u7406\uff0c\u53ef\u4ee5\u5f97\u5230 $PM=PA=PB$\uff0c\u6240\u4ee5 $\\angle APO=\\angle OAP$\u3002\u540c\u6837\u5728\u5927\u76f4\u89d2\u4e09\u89d2\u5f62 $PCD$ \u4e2d\u7c7b\u4f3c\u5730\u53c8\u6709 $\\angle DCP=\\angle OAP$\uff0c\u56e0\u6b64 $\\angle DCP=\\angle APO$\u3002\u540c\u4f4d\u89d2\u76f8\u7b49\uff0c\u4e24\u76f4\u7ebf\u5e73\u884c\u3002<\/p>\n\n\n\n \u8bc1\u6bd5\u3002<\/p>\n\n\n\n **\u62d3\u5c55\u53d8\u5f62**<\/strong>\uff1a\u6839\u636e\u8fd9\u6761\u6027\u8d28\uff0c\u5e7f\u4e49\u5782\u5f84\u5b9a\u7406\u53ef\u4ee5\u63a8\u5e7f\u6210 $k_{OP}\\cdot k_{CD}=-\\frac{b^2}{a^2}$\u3002<\/p>\n\n\n\n ### \u51e0\u4f55\u6027\u8d28 \u5176\u4e09<\/strong><\/p>\n\n\n\n > \u4ece\u8499\u65e5\u5706\u4e0a\u4e00\u70b9 $P$ \u5411\u692d\u5706 $E$ \u5f15\u4e24\u6761\u5207\u7ebf $PA,PB$\uff0c\u5207\u70b9\u4e3a $A,B$\u3002\u90a3\u4e48 $k_{OA}k_{AP}=k_{OB}k_{BP}=-\\frac{b^2}{a^2},k_{OA}k_{OB}=-\\frac{b^4}{a^4}$\u3002<\/p>\n\n\n\n **\u8bc4\u6790**<\/strong>\uff1a\u8fd0\u7528\u5207\u7ebf\u516c\u5f0f\u548c\u5df2\u77e5\u7684\u5782\u76f4\u6761\u4ef6\u5feb\u901f\u89e3\u9898\u3002<\/p>\n\n\n\n **\u8bc1\u660e**<\/strong>\uff1a<\/p>\n\n\n\n \u4ee4 $A(x_1,y_1),B(x_2,y_2)$\uff0c\u5219\u6839\u636e\u5207\u7ebf\u516c\u5f0f\u5f97 $PA:\\frac{x_1}{a^2}x+\\frac{y_1}{b^2}y=1$\uff0c\u659c\u7387\u4e3a $-\\frac{b^2x_1}{a^2y_1}$\uff0c\u4e58\u79ef\u4e3a $-\\frac{b^2x_1}{a^2y_1}\\cdot\\frac{y_1}{x_1}=-\\frac{b^2}{a^2}$\u3002\u540c\u7406\u53ef\u4ee5\u8bc1\u5f97 $k_{OB}k_{PB}=-\\frac{b^2}{a^2}$\u3002<\/p>\n\n\n\n \u7efc\u5408\u4ee5\u4e0a\u4e24\u5f0f $k_{OA}k_{PA}=k_{OB}k_{PB}=-\\frac{b^2}{a^2}$\uff0c\u5f97 $k_{OA}k_{OB}k_{PA}k_{PB}=\\frac{b^4}{a^4}$\uff0c\u6839\u636e\u8499\u65e5\u5706\u7684\u5207\u7ebf\u5782\u76f4\u6761\u4ef6 $k_{PA}k_{PB}=-1$\uff0c\u5f97\u5230 $k_{OA}k_{OB}=-\\frac{b^4}{a^4}$\u3002<\/p>\n\n\n\n \u8bc1\u6bd5\u3002<\/p>\n\n\n\n ## \u963f\u57fa\u7c73\u5fb7\u4e09\u89d2\u5f62<\/strong><\/p>\n\n\n\n \u629b\u7269\u7ebf\u7684\u67d0\u6761\u5f26 $AB$\uff0c\u8fc7 $A,B$ \u7684\u4e24\u6761\u629b\u7269\u7ebf\u7684\u5207\u7ebf\u76f8\u4ea4\u4e8e $P$ \u70b9\uff0c\u4e09\u89d2\u5f62 $PAB$ \u79f0\u4f5c\u8fd9\u4e2a\u629b\u7269\u7ebf\u7684\u963f\u57fa\u7c73\u5fb7\u4e09\u89d2\u5f62\u3002\u5982\u4e0b\u56fe\uff1a<\/p>\n\n\n\n <\/p>\n\n\n\n $\\triangle ABP$ \u548c $\\triangle CDQ$ \u90fd\u662f\u8fd9\u4e2a\u629b\u7269\u7ebf\u7684\u963f\u57fa\u7c73\u5fb7\u4e09\u89d2\u5f62\u3002<\/p>\n\n\n\n \u82e5\u65e0\u7279\u6b8a\u8bf4\u660e\uff0c\u672c\u7ae0\u4e2d\u7684\u629b\u7269\u7ebf $E$ \u5747\u6307\u4ee3\u629b\u7269\u7ebf $y^2=2px(p>0)$\u3002<\/p>\n\n\n\n ### \u51e0\u4f55\u6027\u8d28 \u5176\u4e00<\/strong><\/p>\n\n\n\n > \u963f\u57fa\u7c73\u5fb7\u4e09\u89d2\u5f62\u5728\u629b\u7269\u7ebf\u4e0a\u7684\u5f26\u7684\u4e2d\u70b9\u4e3a $M$\uff0c\u90a3\u4e48\u8be5\u5f26\u6240\u5bf9\u7684\u9876\u70b9 $P$ \u6ee1\u8db3 $PM\/\/x$\u3002<\/p>\n\n\n\n **\u8bc4\u6790**<\/strong>\uff1a\u5de7\u5999\u8fd0\u7528\u5207\u7ebf\u65b9\u7a0b\u89e3\u51b3\u95ee\u9898\u3002<\/p>\n\n\n\n **\u8bc1\u660e**<\/strong>\uff1a<\/p>\n\n\n\n \u4ee4\u5f26\u7684\u7aef\u70b9 $A(x_1,y_1),B(x_2,y_2)$\uff0c\u70b9\u5728\u629b\u7269\u7ebf\u4e0a\u5f97 $x_1=\\frac{y_1^2}{2p},x_2=\\frac{y_2^2}{2p}$\u3002\u6839\u636e\u5207\u7ebf\u65b9\u7a0b\u5f97 $PA:y_1y=px+px_1$\uff0c\u540c\u7406\u5f97 $PB:y_2y=px+px_2$\uff0c\u8054\u7acb\u89e3\u5f97\u4ea4\u70b9 $P(\\frac{y_1y_2}{2p},\\frac{y_1+y_2}{2})$\u3002\u4e2d\u70b9\u5f97 $M(\\frac{y_1^2+y_2^2}{4p},\\frac{y_1+y_2}{2})$\uff0c\u5f97\u5230 $PM\/\/x$\u3002<\/p>\n\n\n\n \u8bc1\u6bd5\u3002<\/p>\n\n\n\n ### \u51e0\u4f55\u6027\u8d28 \u5176\u4e8c<\/strong><\/p>\n\n\n\n > \u5f53\u963f\u57fa\u7c73\u5fb7\u4e09\u89d2\u5f62\u5728\u629b\u7269\u7ebf\u4e0a\u7684\u5f26\u8fc7\u9876\u70b9 $G(x_0,y_0)$ \u65f6\uff0c\u8be5\u5f26\u6240\u5bf9\u9876\u70b9\u7684\u8fd0\u52a8\u8f68\u8ff9\u4e3a $y_0y=p(x+x_0)$\u3002<\/p>\n\n\n\n **\u8bc4\u6790**<\/strong>\uff1a\u5229\u7528\u5207\u70b9\u5f26\u516c\u5f0f\uff0c\u6216\u8005\u662f\u51e0\u4f55\u6027\u8d28\u4e00\u53ef\u4ee5\u8bc1\u660e\u3002\u6b64\u5904\u9009\u7528\u51e0\u4f55\u6027\u8d28\u4e00\u8fdb\u884c\u8bc1\u660e\u3002<\/p>\n\n\n\n **\u8bc1\u660e**<\/strong>\uff1a<\/p>\n\n\n\n \u4ee4\u5e95\u8fb9 $A(x_1,y_1),B(x_2,y_2)$\uff0c\u6839\u636e\u51e0\u4f55\u6027\u8d28\u4e00\u5f97\u9876\u70b9 $P(\\frac{y_1y_2}{2},\\frac{y_1+y_2}{2})$\u3002\u56e0\u4e3a\u5b9a\u70b9 $G$ \u5728 $AB$ \u4e0a\uff0c\u5e94\u6709 $k_{AB}=k_{AG}$\uff0c\u5373\uff1a<\/p>\n\n\n\n $$<\/p>\n\n\n\n \\begin{aligned}<\/p>\n\n\n\n \\dfrac{y_2-y_1}{x_2-x_1}&=\\dfrac{y_1-y_0}{x_1-x_0}<\/p>\n\n\n\n \\\\\\dfrac{y_2-y_1}{\\frac{y_2^2}{2p}-\\frac{y_1^2}{2p}}=\\dfrac{2p}{y_1+y_2}&=\\dfrac{y_1-y_0}{\\frac{y_1^2}{2p}-x_0}<\/p>\n\n\n\n \\\\y_1^2-2px_0&=y_1^2+y_1y_2-y_0(y_1+y_2)<\/p>\n\n\n\n \\\\2px_0&=y_0(y_1+y_2)-y_1y_2<\/p>\n\n\n\n \\\\2px_0&=2y_0y_p-2x_p<\/p>\n\n\n\n \\\\y_0y_p&=p(x_0+x_p)<\/p>\n\n\n\n \\end{aligned}<\/p>\n\n\n\n $$<\/p>\n\n\n\n \u56e0\u6b64 $P$ \u5728\u76f4\u7ebf $y_0y=p(x+x_0)$ \u4e0a\u3002<\/p>\n\n\n\n \u8bc1\u6bd5\u3002<\/p>\n\n\n\n **\u62d3\u5c55\u53d8\u5f62**<\/strong>\uff1a\u6b64\u7ed3\u8bba\u7684\u63a8\u8bba\u6709\u2014\u2014\u5f53\u5e95\u8fb9\u8fc7\u7126\u70b9\u65f6\uff0c\u9876\u70b9\u7684\u8f68\u8ff9\u4e3a\u629b\u7269\u7ebf\u51c6\u7ebf\uff1b\u5e95\u8fb9\u8fc7 $x$ \u8f74\u5b9a\u70b9 $(a,0)$ \u65f6\uff0c\u9876\u70b9\u8f68\u8ff9\u4e3a\u76f4\u7ebf $x=-a$\u3002<\/p>\n\n\n\n ### \u51e0\u4f55\u6027\u8d28 \u5176\u4e09<\/strong><\/p>\n\n\n\n > \u5f53\u963f\u57fa\u7c73\u5fb7\u4e09\u89d2\u5f62\u7684\u5e95\u8fb9\u8fc7\u7126\u70b9\u65f6\uff0c\u963f\u57fa\u7c73\u5fb7\u4e09\u89d2\u5f62\u7684\u9876\u89d2\u4e3a ${90}\\degree$\uff0c\u5373 $PA\\perp PB$\u3002<\/p>\n\n\n\n <\/p>\n\n\n\n **\u8bc4\u6790**<\/strong>\uff1a\u53ef\u4ee5\u501f\u52a9\u51e0\u4f55\u6027\u8d28\u4e00\u6765\u5feb\u901f\u89e3\u51b3\u3002<\/p>\n\n\n\n **\u8bc1\u660e**<\/strong>\uff1a<\/p>\n\n\n\n \u4ee4 $A(x_1,y_1),B(x_2,y_2)$\uff0c\u7531\u51e0\u4f55\u6027\u8d28\u4e00\u53ef\u5f97 $P(\\frac{y_1y_2}{2p},\\frac{y_1+y_2}{2})$\u3002\u4e24\u5207\u7ebf\u659c\u7387\u4e4b\u79ef\u4e3a\uff1a<\/p>\n\n\n\n $$<\/p>\n\n\n\n \\begin{aligned}<\/p>\n\n\n\n k_1k_2&=\\dfrac{y_1-\\frac{y_1+y_2}{2}}{x_1-\\frac{y_1y_2}{2p}}\\times\\dfrac{y_2-\\frac{y_1+y_2}{2}}{x_2-\\frac{y_1y_2}{2p}}<\/p>\n\n\n\n \\\\&=\\dfrac{\\frac{y_1-y_2}{2}}{\\frac{y_1^2-y_1y_2}{2p}}\\times\\dfrac{\\frac{y_2-y_1}{2}}{\\frac{y_2^2-y_1y_2}{2p}}<\/p>\n\n\n\n \\\\&=\\dfrac{p(y_1-y_2)}{y_1(y_1-y_2)}\\times\\dfrac{p(y_2-y_1)}{y_2(y_2-y_1)}<\/p>\n\n\n\n \\\\&=\\dfrac{p^2}{y_1y_2}<\/p>\n\n\n\n \\end{aligned}<\/p>\n\n\n\n $$<\/p>\n\n\n\n \u6700\u540e\u8054\u7cfb\u5230\u629b\u7269\u7ebf\u7126\u70b9\u5f26\u5b9a\u7406\u4e2d $y_1y_2=-p^2$\uff08\u8bbe\u76f4\u7ebf\u4ee3\u5165\u97e6\u8fbe\u5b9a\u7406\u5f97\u51fa\uff09\u53ef\u4ee5\u5f97\u5230\u659c\u7387\u4e4b\u79ef\u4e3a $-1$\uff0c\u5373\u4e24\u76f4\u7ebf\u5782\u76f4\u3002<\/p>\n\n\n\n \u8bc1\u6bd5\u3002<\/p>\n\n\n\n ### \u51e0\u4f55\u6027\u8d28 \u5176\u56db<\/strong><\/p>\n\n\n\n > \u5728\u963f\u57fa\u7c73\u5fb7\u4e09\u89d2\u5f62\u4e2d\uff0c\u6052\u6709 $\\angle PFA=\\angle PFB$\u3002<\/p>\n\n\n\n <\/p>\n\n\n\n **\u8bc4\u6790**<\/strong>\uff1a\u51e0\u4f55\u6cd5\u642d\u914d\u89e3\u6790\u51e0\u4f55\u89e3\u9898\u8f83\u4e3a\u5feb\u901f\u3002<\/p>\n\n\n\n **\u8bc1\u660e**<\/strong>\uff1a<\/p>\n\n\n\n \u8fc7 $A,B$ \u5206\u522b\u4f5c\u51c6\u7ebf\u7684\u5782\u7ebf $AA_1,BB_1$\uff0c\u5782\u8db3\u4e3a $A_1,B_1$\uff0c\u8fde\u63a5 $A_1P,B_1P,A_1F$\uff0c$A_1F\\cap AP=O$\uff0c\u5982\u4e0b\u56fe\uff1a<\/p>\n\n\n\n <\/p>\n\n\n\n \u4ee4 $A(x_1,y_1),B(x_2,y_2)$\uff0c\u6839\u636e\u5207\u7ebf\u516c\u5f0f\u53ef\u5f97 $PA:y=\\frac{p}{y_1}x+\\frac{px_1}{y_1}$\uff0c\u5f97\u5230\u659c\u7387 $k_{PA}=\\frac{p}{y_1}$\u3002\u7531\u5782\u76f4\u5f97 $A_1(-\\frac{p}{2},y_1)$\uff0c\u56e0\u6b64 $A_1F$ \u659c\u7387\u4e3a $-\\frac{y_1}{p}$\uff0c\u4e58\u79ef\u4e3a $-1$\uff0c\u6709 $AP\\perp A_1F$\u3002<\/p>\n\n\n\n \u5728\u629b\u7269\u7ebf\u4e2d\uff0c\u6709 $|AA_1|=|AF|$\uff0c\u6839\u636e\u76f4\u89d2\u4e09\u89d2\u5f62 HL \u578b\u5168\u7b49\u5f97 $\\triangle A_1AO\\cong\\triangle FAO$\uff0c\u8fdb\u800c\u6709 $\\angle A_1AO=\\angle FAO$\uff0c\u518d\u6b21\u53ef SAS \u8bc1\u5f97 $\\triangle A_1AP\\cong\\triangle FAP$\u3002\u4eff\u7167\u4e0a\u8ff0\u5168\u7b49\u63a8\u5bfc\u53ef\u8bc1\u5f97 $\\triangle BFP\\cong\\triangle BB_1P$\u3002\u90a3\u4e48 $\\angle PFB=\\angle BB_1P,\\angle PFA=\\angle PA_1A$\u3002<\/p>\n\n\n\n \u6839\u636e\u51e0\u4f55\u6027\u8d28\u4e00\u53ef\u5f97\uff0c$y_P=\\frac{y_1+y_2}{2}$\uff0c\u5c31\u6709 $A_1P=B_1P$\uff0c$\\angle PA_1B_1=\\angle PB_1A_1$\uff0c\u56e0\u6b64 $\\angle PA_1A=\\angle PB_1B=90\\degree+\\angle PA_1B_1$\uff0c\u8fdb\u800c\u5f97\u5230 $\\angle PFB=\\angle PFA$\u3002<\/p>\n\n\n\n \u8bc1\u6bd5\u3002<\/p>\n\n\n\n ### \u51e0\u4f55\u6027\u8d28 \u5176\u4e94<\/strong><\/p>\n\n\n\n > \u5728\u963f\u57fa\u7c73\u5fb7\u4e09\u89d2\u5f62\u4e2d\uff0c\u6709 $|AF|\\cdot|BF|=|PF|^2$\u3002<\/p>\n\n\n\n **\u8bc4\u6790**<\/strong>\uff1a\u6839\u636e\u6027\u8d28\u4e00\u5f97\u51fa\u7684\u70b9\u7684\u5750\u6807\u4ee3\u5165\u8ba1\u7b97\u5373\u53ef\u9a8c\u8bc1\u3002<\/p>\n\n\n\n **\u8bc1\u660e**<\/strong>\uff1a<\/p>\n\n\n\n \u6839\u636e\u6027\u8d28\u4e00\u53ef\u5f97 $P(\\frac{y_1y_2}{2p},\\frac{y_1+y_2}{2})$\uff0c\u8ddd\u79bb\u516c\u5f0f\u53ef\u5f97 $|PF|^2=(\\frac{y_1y_2}{2p}-\\frac{p}{2})^2+(\\frac{y_1+y_2}{2})^2=\\frac{p^2}{4}+\\frac{y_1^2y_2^2}{4p^2}+\\frac{y_1^2+y_2^2}{4}$\u3002<\/p>\n\n\n\n \u540c\u65f6\uff0c\u5728\u629b\u7269\u7ebf\u4e2d\u6ee1\u8db3 $|AF|=x_A+\\frac{p}{2}=\\frac{y_1^2}{2p}+\\frac{p}{2}$\uff1b\u540c\u7406\u6709 $|BF|=x_B+\\frac{p}{2}=\\frac{y_2^2}{2p}+\\frac{p}{2}$\u3002\u76f8\u4e58\uff1a<\/p>\n\n\n\n $$<\/p>\n\n\n\n \\begin{aligned}<\/p>\n\n\n\n |AF|\\cdot|BF|&=\\left(\\frac{y_1^2}{2p}+\\frac{p}{2}\\right)\\times\\left(\\frac{y_2^2}{2p}+\\frac{p}{2}\\right)<\/p>\n\n\n\n \\\\&=\\frac{y_1^2y_2^2}{4p^2}+\\frac{p^2}{4}+\\frac{y_1^2+y_2^2}{4}<\/p>\n\n\n\n \\\\&=|QF|^2<\/p>\n\n\n\n \\end{aligned}<\/p>\n\n\n\n $$<\/p>\n\n\n\n \u8bc1\u6bd5\u3002<\/p>\n\n\n\n ### \u51e0\u4f55\u6027\u8d28 \u5176\u516d<\/strong><\/p>\n\n\n\n > \u5e95\u8fb9 $AB$ \u957f\u4e3a $a$ \u7684\u963f\u57fa\u7c73\u5fb7\u4e09\u89d2\u5f62\u7684\u9762\u79ef\u6700\u5927\u503c\u4e3a $\\frac{a^3}{8p}$\u3002<\/p>\n\n\n\n **\u8bc4\u6790**<\/strong>\uff1a\u5229\u7528\u4e09\u89d2\u5f62\u9762\u79ef\u7b49\u4e8e\u5e95\u4e58\u9ad8\u9664\u4ee5\u4e8c\uff0c\u518d\u5bf9\u9ad8\u7684\u957f\u5ea6\u8fdb\u884c\u653e\u7f29\u5373\u53ef\u3002<\/p>\n\n\n\n **\u8bc1\u660e**<\/strong>\uff1a<\/p>\n\n\n\n \u5982\u56fe\uff1a$PH$ \u4e3a $\\triangle APB$ \u5728 $AB$ \u8fb9\u4e0a\u7684\u9ad8\uff0c$M$ \u4e3a $AB$ \u4e2d\u70b9\u3002\u4ee4 $AB:x=ky+b$\u3002<\/p>\n\n\n\n <\/p>\n\n\n\n \u6613\u77e5 $|PH|\\leq|PM|$\uff0c\u5728 $AB\\perp x$ \u65f6\u7b49\u53f7\u6210\u7acb\u3002$|AB|=a=\\sqrt{(k^2+1)(y_1-y_2)^2}\\geq\\sqrt{(y_1-y_2)^2}$\u3002<\/p>\n\n\n\n \u6839\u636e\u6027\u8d28\u4e00\uff0c$P(\\frac{y_1y_2}{2p},\\frac{y_1+y_2}{2})$\uff0c$M(\\frac{x_1+x_2}{2},\\frac{y_1+y_2}{2})$\uff0c$|PM|=\\frac{x_1+x_2}{2}-\\frac{y_1y_2}{2p}=\\frac{y_1^2+y_2^2}{4p}-\\frac{y_1y_2}{2p}=\\frac{(y_1-y_2)^2}{4p}$\u3002<\/p>\n\n\n\n \u6b64\u65f6 $S_{\\triangle APB}\\leq\\frac{1}{2}a\\frac{(y_1-y_2)^2}{4p}\\leq\\frac{a^3}{8p}$\uff0c\u5f53\u4e14\u4ec5\u5f53 $AB\\perp x$ \u65f6\u53d6\u5f97\u7b49\u53f7\u3002<\/p>\n\n\n\n \u8bc1\u6bd5\u3002<\/p>\n\n\n\n ## \u65b0\u5b9a\u4e49\u66f2\u7ebf<\/strong><\/p>\n\n\n\n ### \u4f2f\u52aa\u5229\u53cc\u7ebd\u7ebf<\/strong><\/p>\n\n\n\n > \u5e73\u9762\u5185\u4e00\u70b9 $P$ \u5230 $x$ \u8f74\u4e24\u5b9a\u70b9 $(\\pm a,0)$ \u7684\u8ddd\u79bb\u4e4b\u79ef\u4e3a\u5b9a\u503c $a^2$ \u7684\u66f2\u7ebf\u53eb\u505a\u4f2f\u52aa\u5229\u53cc\u7ebd\u7ebf\uff08\u7b80\u79f0\u53cc\u7ebd\u7ebf\uff09\uff0c\u5176\u89e3\u6790\u5f0f\u4e3a $(x^2+y^2)^2=2a^2(x^2-y^2)$\u3002\u82e5\u5b9a\u70b9\u5728 $y$ \u8f74\u4e0a\u5219\u4e3a $(x^2+y^2)^2=2a^2(y^2-x^2)$\u3002<\/p>\n\n\n\n \u4f2f\u52aa\u5229\u53cc\u7ebd\u7ebf $(x^2+y^2)^2=18(x^2-y^2)$ \u7684\u56fe\u50cf\u5982\u4e0b\uff1a<\/p>\n\n\n\n <\/p>\n\n\n\n #### \u8f68\u8ff9\u65b9\u7a0b<\/strong><\/p>\n\n\n\n **\u8bc4\u6790**<\/strong>\uff1a\u8bbe\u70b9\u8ba1\u7b97\u3002<\/p>\n\n\n\n **\u89e3**<\/strong>\uff1a<\/p>\n\n\n\n \u4ee4 $P(x,y),F_1(-a,0),F_2(a,0)$\uff0c$|PF_1||PF_2|=a^2$\uff0c\u53ef\u5f97\uff1a<\/p>\n\n\n\n $$<\/p>\n\n\n\n \\begin{aligned}<\/p>\n\n\n\n |PF_1||PF_2|&=a^2<\/p>\n\n\n\n \\\\\\sqrt{(x+a)^2+y^2}\\sqrt{(x-a)^2+y^2}&=a^2<\/p>\n\n\n\n \\\\\\sqrt{x^2+y^2+a^2-2ax}\\sqrt{x^2+y^2+a^2+2ax}&=a^2<\/p>\n\n\n\n \\\\\\sqrt{x^4+y^4+a^4+2x^2y^2+2x^2a^2+2y^2a^2-4x^2a^2}&=a^2<\/p>\n\n\n\n \\\\\\sqrt{x^4+y^4+a^4+2x^2y^2-2x^2a^2+2y^2a^2}&=a^2<\/p>\n\n\n\n \\\\x^4+y^4+2x^2y^2&=2x^2a^2-2y^2a^2<\/p>\n\n\n\n \\\\(x^2+y^2)^2&=2a^2(x^2-y^2)<\/p>\n\n\n\n \\end{aligned}<\/p>\n\n\n\n $$<\/p>\n\n\n\n **\u62d3\u5c55\u53d8\u5f62**<\/strong>\uff1a\u5b9a\u70b9\u5728 $y$ \u8f74\u4e0a\u65f6\u540c\u7406\u3002<\/p>\n\n\n\n