5. Given that the modulus of the complex number $z$ is 1, then the minimum value of $|z-4|^{2}+|z+3i|^{2}$ is $\qquad$
Answer 17.
Analysis In the complex plane, let $A(4,0), B(0,-3), P$ be a point on the unit circle. The problem is to find the minimum value of $|P A|^{2}+|P B|^{2}$. Let $P(\cos \alpha, \sin \alpha)$, where $\alpha \in \mathbf{R}$.
$$
\begin{aligned}
|P A|^{2}+|P B|^{2} & =(\cos \alpha-4)^{2}+\sin ^{2} \alpha+\cos ^{2} \alpha+(\sin \alpha+3)^{2} \\
& =27+6 \sin \alpha-8 \cos \alpha=27+10 \sin (\alpha+\varphi) \\
& \geq 17
\end{aligned}
$$

Since $\alpha \in \mathbf{R}$, there must exist an $\alpha$ such that $\sin (\alpha+\varphi)=-1$, meaning the equality can be achieved.