已知椭圆些x^2/2+y^2=1过点A(2,1)的直线与椭圆交点M、N,求弦MN中点轨迹方程
问题描述：
已知椭圆些x^2/2+y^2=1过点A(2,1)的直线与椭圆交点M、N,求弦MN中点轨迹方程
答
MN的中点P(x,y)
xM+xN=2x
yM+yN=2y
过点A(2,1)的直线与椭圆交点M、N:
kMN=(y-1)/(x-2)=(yM-yN)/(xM-xN)
x^2/2+y^2=1
(xM)^2/2+(yM)^2=1.(1)
(xN)^2/2+(yN)^2=1.(2)
(1)-(2):
[(xM)^2-(xN)^2]/2+(yM)^2-(yN)^2=0
(xM+xN)*(xM-xN)+2(yM+yN)*(yM-yN)=0
(xM+xN)+2(yM+yN)*(yM-yN)/(xM-xN)=0
2x+2y*(y-1)/(x-2)=0
x(x-2)+y(y-1)=0
(x-1)^2+(y-0.5)^2=1.25