當(dāng)弦PQ不垂直X軸時(shí),設(shè)PQ方程為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
設(shè)根為x1,x2,則(x1-p)^2=2px1/k^2,(x2-p)^2=2px2/k^2
且x1+x2=2p(1+k^2)/k^2,x1x2=p^2
則P(x1,y1),Q(x2,y2)
則MP^2=(根號(hào)(1+k^2))*(x1-p)^2
MQ^2=(根號(hào)(1+k^2))*(x2-p)^2
則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
當(dāng)PQ垂直X軸時(shí),則P(p,p根號(hào)2),Q(p,-p根號(hào)2)
MP^2=2P^2,MQ^2=2P^2則1/MP^2+1/MQ^2=1/P^2
祝你學(xué)習(xí)愉快