Question

已知抛物线y^2=4x,焦点为F,顶点为0,点P在抛物线上移动,M是FP的中点,求点M的轨迹方程.

Answer 1

抛物线y^2=4x,焦点为F(1,0),顶点(0,0)

p点(x,y)

M点[(1+x)/2,y/2]

[(1+y^2/4)/2,y/2]

设M点的轨迹方程为

Y^2=2pX

代入M点

y^2/4=2p(1+y^2/4)/2

y^2=4p(1+y^2/4)

p=y^2/(4+y^2)又y^2=4x

p=4x/(4+4x)=x/(1+x)

M点的运动轨迹方程式y^2=2x^2/(1+x)