AB是抛物线y平方=2px的焦点弦,且AB的模等于m,O是抛物线的顶点 求三角形AOB的面积.
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设,点A坐标为(t1^2/2p,t1),点B坐标为(t2^2/2p,t2),
抛物线y^2=2px,则焦点坐标为(P/2,0).
令,直线AB的方程为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的高为h,
利用点到直线间的距离公式,得
h=|-(p/2)*√[2p/(a-3p)]|/√[2p/(a-3p)+1]
=(p/2)*√[2p/(a-p)],
则三角形AOB的面积是=1/2*AB*h
=(ap/4)*√[2p/(a-p)].
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