已知l为抛物线y2=2px(p>0)的准线,AB为过焦点F的弦,M为AB中点,过M做直线L的垂线,垂足为N交抛物线与点P

设A=(x1^2/2p,x1), B(x2^2/2p,x2)
则AB连线方程为y=2px/(x1+x2)+x1x2/(x1+x2)
过点F(p/2,0)
所以p^2+x1x2=0
p^2=-x1x2

M=[(x1^2+x2^2)/4p,(x1+x2)/2]
N=[-p/2,(x1+x2)/2]
MN的中点为[(x1^2+x2^2-2p^2)/8p,(x1+x2)/2]
2p[(x1^2+x2^2-2p^2)/8p]=(1/4)[x1^2+x2^2+2x1x2]=(1/4)(x1+x2)^2
[(x1+x2)/2]^2=(1/4)(x1+x2)^2
所以MN中点在抛物线上，即P
所以P平分MN