椭圆x2/25＋y2/9＝1,p1,p2,p为该椭圆上任意三点.

题目

椭圆x2/25＋y2/9＝1,p1,p2,p为该椭圆上任意三点.
椭圆x2/25＋y2/9＝1,p1,p2,p为该椭圆上任意三点,且线段p1p2经过椭圆中心O,若直线pp1,pp2的斜率存在且分别为k1,k2,求证：k1*k2＝-9/25.

答案

设P1(x,y),P2(-x,-y) P(m,n)
x^2/25＋y^2/9＝1 (1) ,
m^2/25+n^2/9=1 (2)
(1)-(2):
(x^2-m^2)/25+(y^2-n^2)/9=0
∴(y^2-n^)/(x^2-m^2)=-9/25
k1=(y-n)/(x-m),k2=(-y-n)/(-x-m)
k1*k2=(y-n)(y+n)/(x-m)(x+m)=(y^2-n^)/(x^2-m^2)=-9/25