5.3.1 * Given two vertices of a parallelogram $A(0,0), B(4,-3)$, point $P$ internally divides diagonal $A C$ in the ratio $2: 1$. When point $D$ moves on the circumference of a circle centered at $A$ with a radius of 3, find the locus of point $P$.
Given the figure, let $B, C, D, P$ correspond to the complex numbers $z_{B}, z_{C}, z_{D}, z_{P}$, then
$$
\left\{\begin{array}{l}
z_{C}=z_{B}+z_{D}, \\
z_{C}=\frac{3}{2} z_{P} .
\end{array}\right.
$$

From (1), we get $z_{D}=z_{C}-z_{B}$. By $\left|z_{D}\right|=3$ and (1), (2), we know $\left|\frac{3}{2} z_{P}-z_{B}\right|=3$. Therefore,
$$
\left|z_{P}-\frac{2}{3} z_{B}\right|=2 \text{. }
$$

Substituting $z_{B}=4-3 \mathrm{i}$ into (3), we get $\left|z_{P}-\left(\frac{8}{3}-2 \mathrm{i}\right)\right|=2$. This indicates that the required locus is a circle with center at $\left(\frac{8}{3},-2\right)$ and radius length of 2.