1.設(shè)直線AB的斜率為k (a為直線AB的傾斜角)
當(dāng)a=π/2時,AB垂直于x軸,x=p/2
得y=±p
所以A B的坐標(biāo)分別為(p/2,p),(p/2,-p)
y1*y2=-p^2,x1*x2=p^2/4
當(dāng)a≠π/2
y^2=2px
焦點(p/2,0),準(zhǔn)線x=-p/2
則直線AB：y=k(x-p/2)
拋物線：y^2=2px
聯(lián)立
k^2x^2-(k^2p+2p)x+k^2*p^2/4=0
則x1*x2=p^2/4
y1*y2=-p^2
2.因為拋物線上任一點到焦點的距離等于其到準(zhǔn)線的距離
所以|AB|=|AF|+|B=x1+P/2+x2+P/2=x1+x2+P
(1)當(dāng)a=π/2時,AB垂直于x軸
令x=p/2得y=±p
所以A B的坐標(biāo)分別為(p/2,p),(p/2,-p)
所以弦長AB=2p
(2)當(dāng)a≠π/2時,焦點弦AB的的斜率為k=tana
所以直線為y-0=k(x-p/2)
帶入拋物線y^2=2px(p>0)得k^2(x-p/2)^2=2px
化簡得k^2x^2-(k^2p+2p)x+k^2p^2/4=0
所以x1+x2=(k^2p+2p)/k^2,x1*x2=p^2/4
所以弦長|AB|=√(1+k^2)*|x1-x2|=√(1+k^2)*√[(x1+x2)^2-4x1*x2]
=2(k^2+1)p/k^2=2p(tana^2+1)/tana^2 =2p/(sina)^2