已知椭圆C:x눀/a눀+y눀/b눀=1(a>b>0)经过点(0,1),离心率e=√3/2

2024-12-05 05:47:29
推荐回答(1个)
回答1:

(1) 椭圆过点(0,1),∴b=1,
离心率c/a=√3/2,
∴(c/a)^2=3/4=(a^2-1)/a^2,a^2=4.
∴椭圆方程是x^2/4+y^2=1.①

(2)把x=my+1②代入①,m^2y^2+2my+1+4y^2=4,
整理得(m^2+4)y^2+2my-3=0,
设A(x1,y1),B(x2,y2),
则y1+y2=-2m/(m^2+4),y1y2=-3/(m^2+4),
A'(x1,-y1),A'B的斜率=(y2+y1)/(x2-x1)=(y2+y1)/(my2-my1)(由②),
A'B的方程是y-y2=(y2+y1)(x-x2)/(my2-my1),

令y=0得x=x2-m(y2^2-y1y2)/(y2+y1)=my2+1-m(y2^2-y1y2)/(y2+y1)
=1+m[y2^2+y1y2-y2^2+y1y2]/(y1+y2)
=1+2my1y2/(y1+y2)=1+2m(-3)/(-2m)=4,
∴A'B与x轴交于定点(4,0).
参考:http://zhidao.baidu.com/question/710903105647765365.html