(4) Let the hyperbola $\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1(a>0, b>0)$ have its right focus at $F$. Draw a line $l$ through point $F$ perpendicular to the $x$-axis, intersecting the two asymptotes at points $A$ and $B$. Let $P$ be one of the intersections of $l$ with the hyperbola. Let $O$ be the origin. If there exist real numbers $m$ and $n$ such that $\overrightarrow{O P}=m \overrightarrow{O A}+ n \overrightarrow{O B}$, and $m n=\frac{2}{9}$, then the eccentricity of the hyperbola is ( ).
(A) $\frac{3 \sqrt{2}}{4}$
(B) $\frac{9}{8}$
(C) $\frac{3 \sqrt{5}}{5}$
(D) $\frac{3 \sqrt{2}}{2}$
(4) A Hint: Substitute $A\left(c, \frac{b c}{a}\right), B\left(c,-\frac{b c}{a}\right)$ into
$$
\overrightarrow{O P}=m \overrightarrow{O A}+n \overrightarrow{O B}
$$

to get $P\left((m+n) c,(m-n) \frac{b c}{a}\right)$, substitute into the hyperbola equation, to get
$$
\frac{(m+n)^{2} c^{2}}{a^{2}}-\frac{(m-n)^{2} b^{2} c^{2}}{a^{2} b^{2}}=1,
$$

simplifying gives $4 m n c^{2}=a^{2}$,
$$
e=\frac{c}{a}=\frac{1}{\sqrt{4 m n}}=\sqrt{\frac{9}{8}}=\frac{3 \sqrt{2}}{4} .
$$