設(shè),點(diǎn)A坐標(biāo)為(t1^2/2p,t1),點(diǎn)B坐標(biāo)為(t2^2/2p,t2),
拋物線y^2=2px,則焦點(diǎn)坐標(biāo)為(P/2,0).
令,直線AB的方程為Y=K(X-P/2),
X=(Y+PK/2)/K=(2Y+PK)/(2K).
K=(t2-t1)/[(t2^2-t1^2)/2p]
=2p/(t2+t1).
Y^2=2P*(2Y+PK)/(2K),
ky^2-2py-kp^2=0,
t1+t2=2p/k,
t1*t2=-p^2,
AB=a=x1+x2+p=(t1^2+t2^2)/2p+p=[(t1+t2)^2-4t1*t2]/2p+p,
2p/k^2=a-3p,
k=√[2p/(a-3p)],
y=√[2p/(a-3p)]x-(p/2)*√[2p/(a-3p)],
√[2p/(a-3p)]x-y-(p/2)*√[2p/(a-3p)]=0.
令,三角形AOB的高為h,
利用點(diǎn)到直線間的距離公式,得
h=|-(p/2)*√[2p/(a-3p)]|/√[2p/(a-3p)+1]
=(p/2)*√[2p/(a-p)],
則三角形AOB的面積是=1/2*AB*h
=(ap/4)*√[2p/(a-p)].