曲线系的椭圆系与双曲线系
\u5171\u7126\u70b9\u7684\u692d\u5706\u7cfb,\u5171\u6e10\u8fd1\u7ebf\u7684\u53cc\u66f2\u7ebf\u7cfb\u3002\u5171\u7126\u70b9\u7684\u692d\u5706\u7cfb\uff1ax^2/(a^2+m)+y^2/(b^2+m)=1
\u5171\u6e10\u8fd1\u7ebf\u7684\u53cc\u66f2\u7ebf\u7cfb\uff1aX^2/a^2-Y^2/b^2=M
\uff081\uff09 \u5f530<a<1\uff0c\u65b9\u7a0bax\u5e73\u65b9+y\u5e73\u65b9\u8868\u793a\u7684\u66f2\u7ebf\u662f\uff08\uff09\uff1f
A\u5706 B\u7126\u70b9\u5728X\u8f74\u4e0a\u7684\u692d\u5706 C\u7126\u70b9\u5728Y\u8f74\u7684\u692d\u5706 D\u53cc\u66f2\u7ebf
Ax^2+y^2=1, \u9009\u62e9B
\uff082\uff09 \u692d\u5706\u7684\u4e2d\u5fc3\u5728\u539f\u70b9\uff0c\u6709\u4e00\u4e2a\u7126\u70b9F\uff08o,-1\uff09,\u5b83\u7684\u79bb\u5fc3\u7387\u662f\u65b9\u7a0b2X\u5e73\u65b9-5X+2=0\u7684\u4e00\u4e2a\u6839\uff0c\u692d\u5706\u7684\u65b9\u7a0b\u662f\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u3002
2X^2-5X+2=0==>x1=2,x2=1/2
\u2234e=1/2==>c=2a==>a=2==>b^2=a^2-c^2=3
\u2234x^2/3+y^2/4=1
\uff083\uff09 \u4ee5\u692d\u5706\u77ed\u8f74\u4e3a\u76f4\u5f84\u7684\u5706\u7ecf\u8fc7\u6b21\u692d\u5706\u7684\u7126\u70b9\uff0c\u5219\u692d\u5706\u7684\u79bb\u5fc3\u7387\u662f\u2014\u2014\u2014\u2014\u2014\u2014\u3002
\u7531\u9898\u610fb=c
A^2=b^2+c^2=2c^2,\u2234e=c/a=c/\u221a2c=\u221a2/2
\uff084\uff09 \u692d\u5706\u77ed\u8f74\u7684\u4e00\u4e2a\u7aef\u70b9\u770b\u957f\u8f74\u4e24\u7aef\u70b9\u7684\u89c6\u89d2\u4e3a120\u00b0\uff0c\u5219\u8fd9\u4e2a\u692d\u5706\u7684\u79bb\u5fc3\u7387\u662f\u2014\u2014\u2014\u2014\u2014\u2014\u3002
\u7531\u9898\u610ftan60\u00b0=a/b==>a=\u221a3b
\u2234c=\u221a(3b^2-b^2)= \u221a2b
\u2234e=c/a=\u221a6/3
\uff085\uff09 \u5df2\u77e5F1\u3001F2\u662f\u692d\u5706X\u5e73\u65b9/4 +y\u5e73\u65b9=1\u7684\u4e24\u4e2a\u7126\u70b9\uff0cP\u662f\u8be5\u692d\u5706\u4e0a\u7684\u4e00\u4e2a\u52a8\u70b9\uff0c\u6c42|PF1|*|PF2|\u7684\u6700\u5927\u503c\u2014\u2014\u2014\u2014\u2014\u2014\u3002
\u692d\u5706X^2/4 +y^2=1
|PF1|+|PF2|=4==>|PF1|=4-|PF2|
|PF1|*|PF2|=(4-|PF2|)|PF2|=-|PF2|^2+4|PF2|=(|PF2|-2)^2+4
\u2234|PF1|*|PF2|\u7684\u6700\u5927\u503c4
\u8bbeF1\u3001F2\u5206\u522b\u4e3a\u692d\u5706C:X\u5e73\u65b9/a\u5e73\u65b9\u5e73\u65b9+Y/b\u5e73\u65b9=1\uff08a>b>0\uff09\u7684\u5de6\u3001\u53f3\u4e24\u4e2a\u7126\u70b9\uff0c\uff081\uff09\u82e5\u692d\u5706C\u4e0a\u7684\u70b9A\uff081\uff0c3/2\uff09\u5230F1,F2\u4e24\u70b9\u7684\u8ddd\u79bb\u4e4b\u548c\u7b49\u4e8e4\uff0c\u5199\u51fa\u692d\u5706C\u7684\u65b9\u7a0b\uff1b\uff082\uff09\u8bbeK\u662f\uff081\uff09\u4e2d\u6240\u5f97\u692d\u5706\u7684\u52a8\u70b9\uff0c\u6c42\u7ebf\u6bb5F1K\u7684\u4e2d\u70b9\u7684\u8f68\u8ff9\u65b9\u7a0b\uff1b
(1)\u89e3\u6790\uff1a\u2235\u692d\u5706C:X^2/a^2+Y^2/b^2=1\uff08a>b>0\uff09, |AF1|+|AF2|=4,A(1,3/2)
\u2234\u221a[(1+c)^2+(3/2)^2]+\u221a[(1-c)^2+(3/2)^2]=4\uff0c\u89e3\u5f97c=1
A=2==a^2=4,b^2=3
\u2234\u692d\u5706C\u7684\u65b9\u7a0bX^2/4+Y^2/3=1
(2)\u89e3\u6790\uff1aK(x0,y0), \u7ebf\u6bb5F1K\u7684\u4e2d\u70b9P(x,y),F1(-1,0)
2x=x0-1==>x0=2x+1,2y=y0/2==>y0=4y
\u2234(2x+1)^2/4+16y^2/3=1
\u7ebf\u6bb5F1K\u7684\u4e2d\u70b9\u7684\u8f68\u8ff9\u65b9\u7a0b\u4e3a(x+1/2)^2+y^2/(3/16)=1
概念:具有某种共同属性的椭圆或双曲线的集合,称为椭圆系或双曲线系。
几种常见的椭圆系或双曲线系方程:
(1)x^2/(c^2+t)+y^2/t=1(半焦距为c且c≠0),当t>0时,表示共焦点(±c,0)的椭圆系;当-c^2<t<0时,表示共焦点(±c,0)的双曲线系,其他情况无轨迹。
(2)与椭圆或双曲线x^2/a^2±y^2/b^2=1具有相同离心率的椭圆系或双曲线系方程为x^2/a^2±y^2/b^2=λ(λ>0)。
(3)与椭圆x^2/a^2+y^2/b^2=1(a^2>b^2)共焦点的曲线系方程可设为x^2/(a^2-λ)+y^2/(b^2-λ),当λ<b^2时,方程表示与以上椭圆共焦点的椭圆系,当b^2<λ<a^2时,方程表示与以上椭圆共焦点的双曲线系。
(4)渐近线方程为x/a±y/b=1或y=±(b/a)x的双曲线系可设为x^2/a^2-y^2/b^2=λ(λ≠0)。
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