已知點\(F\)為拋物線\(y^2=2px(p>0)\)的焦點\(,\)經過點\(F\)且傾斜角為\(\alpha(0<\alpha<\frac{\pi}{2})\)的直線與拋物線相
交於\(A,B\)兩點\(,\)\(\triangle OAB(O\)為坐標原點\()\)的面積為\(2\sin^3\alpha\)\(,\)線段\(AB\)的垂直平分線與\(x\)軸相交於點\(M\).
則\(|FM|\)的值為\(\underline{\qquad\blacktriangle\qquad}.\)
(晚上配圖）由題意可知，\(S_{\triangle OAB}=\frac{1}{2}|OF|\cdot|AB|\cdot\sin\alpha\)
\(\Rightarrow p^2=4\sin^4\alpha\)
記\(AB\)中點為\(D\),則\(|FD|=\frac{|AF|-|BF|}{2}=\frac{p\cos\alpha}{1-\cos^2\alpha}\)
\(\Rightarrow |FM|=\frac{|FD|}{\cos\alpha}=2\)