已知点(F)为抛物线(y^2=2px(p>0))的焦点(,)经过点(F)且倾斜角为(alpha(0<alpha<frac{pi}{2}))的直线与抛物线相
交于(A,B)两点(,)( riangle OAB(O)为坐标原点())的面积为(2sin^3alpha)(,)线段(AB)的垂直平分线与(x)轴相交于点(M).
则(|FM|)的值为(underline{qquadlacktriangleqquad}.)
(晚上配图）由题意可知，(S_{ riangle OAB}=frac{1}{2}|OF|cdot|AB|cdotsinalpha)
(Rightarrow p^2=4sin^4alpha)
记(AB)中点为(D),则(|FD|=frac{|AF|-|BF|}{2}=frac{pcosalpha}{1-cos^2alpha})
(Rightarrow |FM|=frac{|FD|}{cosalpha}=2)