成都市2020届三诊16题

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\(,\)\(\triangle OAB(O\)为坐标原点\()\)的面积为\(2\sin^3\alpha\)\(,\)线段\(AB\)的垂直平分线与\(x\)轴相交于点\(M\).
则\(|FM|\)的值为\(\underline{\qquad\blacktriangle\qquad}.\)

(晚上配图）由题意可知，\(S_{\triangle OAB}=\frac{1}{2}|OF|\cdot|AB|\cdot\sin\alpha\)

\(\Rightarrow p^2=4\sin^4\alpha\)

记\(AB\)中点为\(D\),则\(|FD|=\frac{|AF|-|BF|}{2}=\frac{p\cos\alpha}{1-\cos^2\alpha}\)

\(\Rightarrow |FM|=\frac{|FD|}{\cos\alpha}=2\)