已知过抛物y^2=2px的焦点F的直线交抛物线于A(X1,Y1)B（X2,Y2)（1）求证X1X2为定值（2）1/FA+1/FB为定值

已知过抛物y^2=2px的焦点F的直线交抛物线于A(X1,Y1)B（X2,Y2)（1）求证X1X2为定值（2）1/FA+1/FB为定值
已知过抛物y^2=2px的焦点F的直线交抛物线于A(X1,Y1)B（X2,Y2)（1）求证X1X2为定值（2）1/FA+1/FB为定值

已知过抛物y^2=2px的焦点F的直线交抛物线于A(X1,Y1)B（X2,Y2)（1）求证X1X2为定值（2）1/FA+1/FB为定值
1)
AB直线方程为：y=k(x-p/2)
代人：y^2=2px得：
k^2(x-p/2)^2=2px
k^2x^2-(pk^2-2p)x+k^2p^2/4=0
x1x2=p^2/4
是定值
2）
FA^2=(x1-p/2)^2+y1^2
=x1^2-px1+p^2/4+2px1
=x1^2+px1+p^2/4
=(x1+p/2)^2
FA=x1+p/2
同样可得：FB=x2+p/2
1/FA+1/FB
=1/(x1+p/2)+1/(x2+p/2)
=(x1+x2+p)/(x1+p/2)(x2+p/2)
=(x1+x2+p)/(x1x2+p(x1+x2)/2+p^2/4)
=(x1+x2+p)/(p^2/4+p(x1+x2)/2+p^2/4)
=(x1+x2+p)/(p(x1+x2+p)/2)
=2/p
为定值