\((2)\)作抛物线\(y\)\({\,\!}^{2}\)\(=2px(p > 0)\)交\(C\)\({\,\!}_{1}\)于\(A\)，\(B\)两点，交\(C\)\({\,\!}_{2}\)于\(C\)，\(D\)两点，当以\(A\)，\(B\)，\(C\)，\(D\)四点为顶点的凸四边形面积为最大时，求实数\(p\)的值．

已知“若点\(P(x_{0},y_{0})\)在双曲线\(C\)：\(\dfrac{{{x}^{2}}}{{{a}^{2}}}-\dfrac{{{y}^{2}}}{{{b}^{2}}}=1(a > 0,b > 0)\)上，则\(C\)在点\(P\)处的切线方程为\(\dfrac{{{x}_{0}}x}{{{a}^{2}}}-\dfrac{{{y}_{0}}y}{{{b}^{2}}}=1\)，现已知双曲线\(C\)：\(\dfrac{{{x}^{2}}}{4}-\dfrac{{{y}^{2}}}{12}=1\)和点\(Q(1,t)(t\ne \pm \sqrt{3})\)，过点\(Q\)作双曲线\(C\)的两条切线，切点分别为\(M\)，\(N\)，则直线\(MN\)过定点\((\)    \()\)

已知“若点\(P\left( {{x}_{0}},{{y}_{0}} \right)\)在双曲线\(C\)：\(\dfrac{{{x}^{2}}}{{{a}^{2}}}-\dfrac{{{y}^{2}}}{{{b}^{2}}}=1\left( a > 0,b > 0 \right)\) 上，则\(C\)在点\(P\)处的切线方程为\(\dfrac{{{x}_{0}}x}{{{a}^{2}}}-\dfrac{{{y}_{0}}y}{{{b}^{2}}}=1\)”，现已知双曲线\(C\)：\(\dfrac{{{x}^{2}}}{4}-\dfrac{{{y}^{2}}}{12}=1\)和点\(Q\left( 1,t \right)\left( t\ne \pm \sqrt{3} \right)\)，过点\(Q\)作双曲线\(C\)的两条切线，切点分别为\(M\)，\(N\)，则直线\(MN\)过定点\((\)    \()\)

已知双曲线\( \dfrac{{x}^{2}}{{a}^{2}}- \dfrac{{y}^{2}}{{b}^{2}}=1\left(a > 0,b > 0\right) \)的右焦点为\(F\)，过点\(F\)向双曲线的一条渐近线引垂线，垂足为\(M\)，交另一条渐近线于点\(N\)，若\({2}\overrightarrow{MF}=\overrightarrow{FN}\)，则双曲线的离心率为_______________．

已知\(P\)是焦距为\(4\sqrt{2}\)的双曲线\(C\)：\(\dfrac{{{x}^{2}}}{{{a}^{2}}}-\dfrac{{{y}^{2}}}{{{b}^{2}}}=1\left( a > 0,b > 0 \right)\)上一点，过\(P\)的直线与双曲线\(C\)的两条渐近线分别交于点\({{P}_{1}}\)，\({{P}_{2}}\)，且\(3\overrightarrow{OP}=\overrightarrow{O{{P}_{1}}}+2\overrightarrow{O{{P}_{2}}}\)，\(O\)为坐标原点．

\((\)Ⅰ\()\)设\({{P}_{1}}\left( {{x}_{1}},{{y}_{1}} \right)\)，\({{P}_{2}}\left( {{x}_{2}},{{y}_{2}} \right)\)，证明：\({{x}_{1}}{{x}_{2}}-{{y}_{1}}{{y}_{2}}=9\)；

\((\)Ⅱ\()\)试求当\(\Delta O{{P}_{1}}{{P}_{2}}\)面积取得最大值时双曲线的方程．