AB是抛物线y^2=2px的焦点弦AB=m,O是抛物线的顶点，则三角形AOB面积

A（x1,y1）B(x2,y2) y2<0
F(p/2,0)
AB的直线方程为
y=k(x-p/2)
x=y/k+p/2
三角形AOB的面积=SAOF+SBOF=(1/2)*(P/2)*|y1|+(1/2)*(p/2)*|y2|
=(p/4)*(y1-y2)
y^2=2px=2p(y/k+p/2)
y^2-2py/k-p^2=0
y1+y2=2p/k
y1y2=-p^2
(y1-y2)^2=(y1+y2)^2-4y1y2
=4p^2/k^2+4p^2
=4p^2(1/k^2+1)
=4p^2(k^2+1)/k^2
当k趋向无穷大时,即AB垂直x轴
(k^2+1)/k^2趋向1
y1-y2=2p
S△AOB=p^2/2

=p^2/2