动直线mx+ny=1交椭圆x2+y2=1于M N两点,点O为椭圆的中心,若OM垂直于ON,求m n应满足的条件

题目

动直线mx+ny=1交椭圆x2+y2=1于M N两点,点O为椭圆的中心,若OM垂直于ON,求m n应满足的条件

答案

设M N两点坐标为（x1,y1）、（x2,x2）,由
x^2+y^2=1
mx+ny=1,
可知（n^2+m^2)x^2-2mx+1-n^2=0,x1x2=(1-n^2)/(n^2+m^2)
（n^2+m^2)y^2-2ny+1-m^2=0,y1y2=(1-m^2)/(n^2+m^2)
又(y1/x1)(y2/x2)=-1,即
y1y2+x1x2=0
(1-n^2)/(n^2+m^2)+(1-m^2)/(n^2+m^2)=0
n^2+m^2=2