倾斜角为45度的直线L交椭圆Xˇ2/4+Yˇ2=1于A,B两点,求线段AB中点M的轨迹方程.

AB中点M(x,y)
xA+xB=2x,yA+yB=2y
(yA-yB)/(xA-xB)=k(AB)=tan45°=1
x^2/4+y^2=1
x^2+4y^2=4
(xA)^2+4(yA)^2=4.(1)
(xB)^2+4(yB)^2=4.(2)
(1)-(2):
(xA)^2-(xB)^2+(yA)^2-(yB)^2=0
(xA+xB)*(xA-xB)+4(yA+yB)*(yA-yB)=0
(xA+xB)+4(yA+yB)*(yA-yB)/(xA-xB)=0
2x+4*2y=0
线段AB中点M的轨迹方程:x+4y=0