Example 3 Point $P$ moves on the circle $x^{2}+(y-3)^{2}=\frac{1}{4}$, and point $Q$ moves on the ellipse $x^{2}+4 y^{2}=4$. Find the maximum value of $|P Q|$, and the corresponding coordinates of point $Q$.
Solve As shown in Figure 10-1, obviously, the line connecting such $P$ and $Q$ must pass through the circle center $O^{\prime}$.
Let $Q(2 \cos \alpha, \sin \alpha)$, and $O^{\prime}(0,3)$, then
$\left|()^{\prime} Q\right|^{2}=4 \cos ^{2} \alpha+(\sin \alpha-3)^{2}=-3 \sin ^{2} \alpha-6 \sin \alpha+13=-3(\sin \alpha+1)^{2}+16 \leqslant 16$.
Thus, $\left|O^{\prime} Q\right| \leqslant 4$, at this point, $\sin \alpha=-1, \cos \alpha=0$.
Therefore, the maximum value of $|P Q|$ is $4 \frac{1}{2}$. The corresponding coordinates of point $Q$ are $(0,-1)$.