Question

椭圆x^2+4y^2=16内一点P（1,-1）,求过点P的弦的中点的轨迹方程

Answer 1

P(1,-1),过点P的弦AB,弦AB的中点M(x,y)
xA+xB=2x,yA+yB=2y
k(AB)=(yA-yB)/(xA-xB)=(y+1)/(x-1)
x^2+4y^2=16
(xA)^2+4(yA)^2=16.(1)
(xB)^2+4(yB)^2=16.(2)
(1)-(2):
(xA+xB)*(xA-xB)+4(yA+yB)*(yA-yB)=0
(xA+xB)+4(yA+yB)*(yA-yB)/(xA-xB)=0
2x+4*2y*(y+1)/(x-1)=0
x*(x-1)+4y*(y+1)=0
过点P的弦的中点的轨迹方程椭圆:
(x-0.5)^2+4(y+0.5)^2=1.25