Example 3. Let the ellipse be $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$ with two moving points $P$ and $Q$ such that the angle subtended at the center (i.e., $\angle P O Q=\frac{\pi}{2}$). Find the maximum and minimum values of $\frac{1}{|O P|}+\frac{1}{|O Q|}$.
Given the pole as $O \%$ and the polar axis as $O X$, the coordinates of $P$ and $Q$ can be set as: $P( \rho_1, 0), Q\left(\rho_{2}, \frac{\pi}{2}+\theta\right)$.
In the polar coordinate system:
$$
\begin{array}{l}
P\left(\rho_{1} \cos \theta, \rho_{1} \sin \theta\right), \\
Q\left(\rho_{2} \cos \left(\frac{\pi}{2}+\theta\right), \rho_{2} \sin \left(\frac{\pi}{2}+\theta\right) .\right.
\end{array}
$$

Since $P$ and $Q$ are on the ellipse, we have
$$
\frac{1}{\rho_{1}^{2}}+\frac{1}{\rho_{2}^{2}}=\frac{1}{a^{2}}+\frac{1}{b^{2}} \text {. }
$$

Also, by $\left(\frac{1}{\rho_{1}}+\frac{1}{\rho_{2}}\right)^{2}=\frac{1}{\rho_{1}^{2}}+\frac{1}{\rho_{2}^{2}}+2 \frac{1}{\rho_{1} \rho_{2}}$,
$\therefore$ when $\sin 2 \theta=1$,
$$
\left(\frac{1}{O P}+\frac{1}{O Q}\right)_{\max }=\frac{\sqrt{2\left(a^{2}+b^{2}\right)}}{a b},
$$

and when $\sin 2 \theta=0$,
$$
\left(\frac{1}{O P}+\frac{1}{O Q}\right)_{\min}=\frac{a+b}{a b}
$$
(Author: Liu Nianlai, Hebei Cangzhou No.2 High School, Tian Chengde—Hebei Cangzhou Pipeline School)