Four. (20 points) From a point $M$ on the ellipse $\frac{x^{2}}{9}+\frac{y^{2}}{4}=1$, two tangents are drawn to the circle with the minor axis as its diameter, with points of tangency $A$ and $B$. The line $AB$ intersects the $x$-axis and $y$-axis at points $P$ and $Q$, respectively. Find the minimum value of $|PQ|$.

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Let point $M(3 \cos \alpha, 2 \sin \alpha)$, and the points of tangency be $A\left(x_{1}, y_{1}\right)$ and $B\left(x_{2}, y_{2}\right)$. The equations of the tangent lines $M A$ and $M B$ are
$$
\begin{array}{l}
x_{1} \cdot 3 \cos \alpha + y_{1} \cdot 2 \sin \alpha = 4, \\
x_{2} \cdot 3 \cos \alpha + y_{2} \cdot 2 \sin \alpha = 4.
\end{array}
$$

Thus, the equation of the line $A B$ is
$$
x \cdot 3 \cos \alpha + y \cdot 2 \sin \alpha = 4.
$$

The points $P\left(\frac{4}{3 \cos \alpha}, 0\right)$ and $Q\left(0, \frac{2}{\sin \alpha}\right)$.
Therefore, $|P Q| = \sqrt{\frac{16}{9 \cos ^{2} \alpha} + \frac{4}{\sin ^{2} \alpha}}$
$$
= \sqrt{\frac{52}{9} + \frac{16}{9} \tan ^{2} \alpha + 4 \cot ^{2} \alpha} \geqslant \frac{10}{3}.
$$

Thus, the minimum value of $|P Q|$ is $\frac{10}{3}$. At this time,
$$
\begin{array}{l}
M\left(\frac{3 \sqrt{10}}{5}, \frac{2 \sqrt{15}}{5}\right),\left(\frac{3 \sqrt{10}}{5}, -\frac{2 \sqrt{15}}{5}\right), \\
\left(-\frac{3 \sqrt{10}}{5}, \frac{2 \sqrt{15}}{5}\right),\left(-\frac{3 \sqrt{10}}{5}, -\frac{2 \sqrt{15}}{5}\right).
\end{array}
$$