焦點(diǎn)M(1,0),設(shè)直線x-1=ky,
(之所以這樣設(shè),而不設(shè)y=k(x-1)是因?yàn)檫@樣可以包括垂直于x軸的那條直線x=1的那種情況)
則(x-1)²=x²-2x+1=k²y²=4k²x
x²-(2+4k²)x+1=0
x1+x2=2+4k²,x1x2=1.
線段MA、MB的長(zhǎng)是點(diǎn)A、B到準(zhǔn)線x=-p/2的距離,
則|MA|=x1+1,|MB|=x2+1,
1/|MA|+1/|MB|=1/(x1+1)+1/(x2+1)
=(x1+x2+2)/(x1x2+x1+x2+1)
=4(1+k²)/(1+2+4k²+1)
=1
一般的情況：
焦點(diǎn)F(p/2,0),設(shè)直線x-p/2=ky,
則x²-px+p²/4=k²y²=2pk²x
x²-(p+2pk²)x+p²/4=0
x1+x2=p+2pk²,x1x2=p²/4.
線段FA、FB的長(zhǎng)是點(diǎn)A、B到準(zhǔn)線x=-p/2的距離,
則FA=x1+p/2,FB=x2+p/2,
1/FA+1/FB=1/(x1+p/2)+1/(x2+p/2)
=(x1+x2+p)/[x1x2+p(x1+x2)/2+p²/4]
=2p(1+k²)/[p²/4+p²/2+p²k²+p²/4]
=2p(1+k²)/[p²(1+k²)]
=2/p