12・32 Let $z_{1}=-1+i, \quad z_{2}=2+5 i$, and the complex number $z$ satisfies $\left|z-z_{1}\right|-$ $\left|z-z_{2}\right|=5$. Then the locus of the point representing the complex number $z$ is
(A) a hyperbola.
(B) one branch of a hyperbola.
(C) a straight line.
(D) a ray.
(High School Mathematics Competition in Huanggang Region, Hubei Province, China, 1989)
[Solution] Since $\left|z_{1}-z_{2}\right|=|-3-4 i|$
$$
= \sqrt{3^{2}+4^{2}}=5 \text {, }
$$

and $\quad\left|z-z_{1}\right|-\left|z-z_{2}\right|=5$
represents the set of points whose distance difference to $Z_{1}$ and $Z_{2}$ is 5, which indicates a ray with $Z_{2}$ as the endpoint (with the same slope as the line passing through $Z_{1}, Z_{2}$), as shown in the figure. Therefore, the answer is $(D)$.