当弦PQ不垂直X轴时,设PQ方程为y=k(x-p),代入y^2=2px
得k^2(x-p)^2=2px,整理得k^2x^2-2p(1+k^2)x+p^2k^2=0
设根为x1,x2,则(x1-p)^2=2px1/k^2,(x2-p)^2=2px2/k^2
且x1+x2=2p(1+k^2)/k^2,x1x2=p^2
则P(x1,y1),Q(x2,y2)
则MP^2=(根号(1+k^2))*(x1-p)^2
MQ^2=(根号(1+k^2))*(x2-p)^2
则1/MP^2+1/MQ^2,(代入(x1-p)^2=2px1/k^2,(x2-p)^2=2px2/k^2)
=k^2/(2p(1+k^2))*(x1+x2)/(x1x2)=1/p^2
当PQ垂直X轴时,则P(p,p根号2),Q(p,-p根号2)
MP^2=2P^2,MQ^2=2P^2则1/MP^2+1/MQ^2=1/P^2
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