5. $P$ is a moving point on the ellipse $\frac{x^{2}}{16}+\frac{y^{2}}{9}=1$ in the first quadrant. Two tangent lines $P A$ and $P B$ are drawn from point $P$ to the circle $x^{2}+y^{2}=9$, with points of tangency at $A$ and $B$ respectively. The line $A B$ intersects the $x$-axis and $y$-axis at points $M$ and $N$, respectively. The minimum value of the area of $\triangle M O N$ is $\qquad$
5. Solution: Let $P(4 \cos \pi, 3 \sin \pi), \pi \in\left(0, \frac{\pi}{2}\right)$, then the equation of line $A B$ is $4 x \cos \pi+3 y \sin \pi=9$.
Thus, $|O M|=\frac{9}{4 \cos \pi},|O N|=\frac{3}{\sin \pi}, S_{\triangle M O N}=\frac{1}{2}|O M|$
- $|O N|=\frac{27}{4 \sin 2 \pi} \geqslant \frac{27}{4}$, when $\pi=\frac{\pi}{4}$, i.e., $P$ is $\left(2 \sqrt{2}, \frac{3 \sqrt{2}}{2}\right)$, the equality holds, hence the minimum value of $S_{\triangle M O N}$ is $\frac{27}{4}$.