6. $P$ is a moving point on the ellipse $\frac{x^{2}}{16}+\frac{y^{2}}{9}=1$ in the first quadrant. Two tangent lines $P A, P B$ are drawn from point $P$ to the circle $x^{2}+y^{2}=9$, with the points of tangency being $A, B$. The line $A B$ intersects the $x$-axis and $y$-axis at points $M, N$, respectively. The minimum value of the area of $\triangle M O N$ is . $\qquad$
6. $\frac{27}{4}$.

Let $P(4 \cos \theta, 3 \sin \theta), \theta \in\left(0, \frac{\pi}{2}\right)$, then the equation of line $AB$ is $4 x \cos \theta+3 y \sin \theta=9$. Therefore, $|O M|=\frac{9}{4 \cos \theta},|O N|=\frac{3}{\sin \theta}$,
$$
S_{\triangle M O N}=\frac{1}{2}|O M| \cdot|O N|=\frac{27}{4 \sin 2 \theta} \geqslant \frac{27}{4} .
$$

When $\theta=\frac{\pi}{4}$, i.e., point $P$ is $\left(2 \sqrt{2}, \frac{3 \sqrt{2}}{2}\right)$, the equality holds. Hence, the minimum value of $S_{\triangle M O N}$ is $\frac{27}{4}$.