Four, the trajectory equation of the moving point $Z_{1}$ in the complex plane is $\left|z_{1}-z_{0}\right|=\left|z_{1}\right|, z_{0}$ is a fixed point, $z_{0} \neq 0$, another moving point $z$ satisfies $z_{1} z=-1$, find the trajectory of point $z$, and indicate its shape and position on the complex plane.
From $Z_{1} Z=-1$, we get $Z_{1}=-\frac{1}{Z}$. Substituting into $\left|Z_{1}-Z_{0}\right|=\left|Z_{1}\right|$, we have
$$
\left|\frac{1}{Z}+Z_{0}\right|=\left|\frac{1}{Z}\right|
$$

Multiplying both sides by $\left|\frac{Z}{Z_{0}}\right|$, we get
$$
\left|Z+\frac{1}{Z_{0}}\right|=\frac{1}{\left|Z_{0}\right|} .
$$

This is a circle with center at $-\frac{1}{Z_{0}}$ and radius $\left|\frac{1}{Z_{0}}\right|$,
Also, $Z \neq 0$, otherwise $Z_{1} Z=0 \neq-1$,
Therefore, the required locus is the curve of this circle with the point $Z=0$ removed.