6. $P$ is a moving point on the ellipse $\frac{x^{2}}{16}+\frac{y^{2}}{9}=1$ in the first quadrant. Tangents $P A$ and $P B$ are drawn from point $P$ to the circle $x^{2}+y^{2}=9$, touching the circle at points $A$ and $B$ respectively. The line $A B$ intersects the $x$-axis and $y$-axis at points $M$ and $N$ respectively. Then the minimum value of $S_{\triangle M O N}$ is ( ).
(A) $\frac{9}{2}$
(B) $\frac{9 \sqrt{3}}{2}$
(C) $\frac{27}{4}$
(D) $\frac{27 \sqrt{3}}{4}$
6.C.

Let $P(4 \cos \theta, 3 \sin \theta), \theta \in\left(0, \frac{\pi}{2}\right)$, then the equation of the line $A B$ is
$4 x \cos \theta+3 y \sin \theta=9$.
Thus, $|O M|=\frac{9}{4 \cos \theta},|O N|=\frac{3}{\sin \theta}$,
$S_{\triangle M O N}=\frac{1}{2}|O M| \cdot|O N|=\frac{27}{4 \sin 2 \theta} \geqslant \frac{27}{4}$.
When $\theta=\frac{\pi}{4}$, i.e., point $P$ is $\left(2 \sqrt{2}, \frac{3 \sqrt{2}}{2}\right)$, the equality holds.
Therefore, the minimum value of $S_{\triangle M O N}$ is $\frac{27}{4}$.