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两条动直线 ${\displaystyle y=k_{1}x}$ 和 ${\displaystyle y=k_{2}x}$ 分别与抛物线 ${\displaystyle C:y^2=2px(p>0)}$ 相交于不同于原点的 ${\displaystyle A,B}$ 两点，当 ${\displaystyle \mathrm{\Delta }OAB}$ 的垂心恰是 ${\displaystyle C}$ 的焦点时，${\displaystyle |AB|=4\sqrt{5}}$.

1. 求 ${\displaystyle p}$;
2. 若 ${\displaystyle k_1{k}_2=-4}$, 弦 ${\displaystyle AB}$ 中点为 ${\displaystyle P}$, 点 ${\displaystyle M(-2,0)}$ 关于直线 ${\displaystyle AB}$ 的对称点 ${\displaystyle N}$ 在抛物线 ${\displaystyle C}$ 上，求 ${\displaystyle \mathrm{\Delta }PMN}$ 的面积.

1. 若 ${\displaystyle \mathrm{\Delta }OAB}$ 的垂心为焦点 ${\displaystyle F{\displaystyle (\frac{p}{2},0)}}$, 则 ${\displaystyle OF\perp AB}$，${\displaystyle |y_A|=|y_B|=2\sqrt{5}}$, 从而 ${\displaystyle x_A=x_B={\displaystyle \frac{10}{p}}}$. 由于 ${\displaystyle AF\perp OB}$, 故$$\begin{array}{r}0=\overrightarrow{OB}\cdot \overrightarrow{FA}=x_1(x_1-\frac{p}{2})-y_1^2={\left(\frac{10}{p}\right)}^2-5-20\Rightarrow p=2.\end{array}$$
2. 联立 ${\displaystyle y^2=4x}$ 与 ${\displaystyle y=k_{1}x,y=k_{2}x}$，解得 ${\displaystyle A{\displaystyle (\frac{4}{k_1^2},\frac{4}{k_1}),B(\frac{4}{k_2^2},\frac{4}{k_2})}}$, 则 ${\displaystyle AB}$ 的横截距为$$\begin{array}{r}\frac{x_A{y}_B-x_B{y}_A}{y_B-y_A}=\frac{\frac{4}{k_1^2}\cdot \frac{4}{k_2}-\frac{4}{k_2^2}\cdot \frac{4}{k_1}}{\frac{4}{k_2}-\frac{4}{k_1}}=-\frac{4}{k_1{k}_2}=1.\end{array}$$
   也即 ${\displaystyle AB}$ 恒过 ${\displaystyle F(1,0)}$. 由于 ${\displaystyle N,M}$ 关于 ${\displaystyle AB}$ 对称，故 ${\displaystyle NF=MF=3}$, 即 $$\begin{array}{r}9=y_N^2+(x_N-1{)}^2=4x_N+(x_N-1{)}^2\Rightarrow x_N=2(\text{舍去负值})\end{array}$$
   也即 ${\displaystyle N{\displaystyle (2,\pm 2\sqrt{2})}}$, 不妨设是 ${\displaystyle (2,2\sqrt{2})}$. 则 ${\displaystyle l_{MN}:y={\displaystyle \frac{\sqrt{2}}{2}(x+2)}}$, 而 ${\displaystyle M,N}$ 中点为 ${\displaystyle (0,\sqrt{2})}$, 故 ${\displaystyle l_{AB}:y=-\sqrt{2}(x-1)}$, 与 ${\displaystyle C}$ 联立得 ${\displaystyle x^2-4x+1=0}$, 也即 ${\displaystyle x_P=2,P(2,-\sqrt{2})}$. 故由点到直线距离公式，${\displaystyle P}$ 到 ${\displaystyle MN}$ 的距离为 ${\displaystyle d=2\sqrt{3}}$, 故 ${\displaystyle S_{\mathrm{\Delta }PMN}}$ 的面积为 ${\displaystyle {\displaystyle \frac{1}{2}MN\cdot d=6\sqrt{2}}}$.