Example 1. $O$ is the origin of the complex plane, $Z_{1}$ and $Z_{2}$ are moving points, $\theta$ and $-\theta\left(0<\theta<\frac{\pi}{2}\right)$;
( 2 ) The area of $\triangle O Z_{1} Z_{2}$ is a constant $S$. Find the minimum multiple of the modulus of the complex number corresponding to the centroid of $\triangle O Z_{1} Z_{1}$.
Let the coordinates of $Z_{1}, Z_{2}, Z$ be $\left(\gamma_{1} \cos \theta, \gamma_{1} \sin \theta\right)$, $\left(\gamma_{2} \cos (-\theta), \gamma_{2} \sin (-\theta)\right)$, $(x, y)$, and $Z$ be the centroid of $\triangle O Z_{1} Z_{2}$,
$$
x=\frac{1}{3}\left(\gamma_{1}+\gamma_{2}\right) \cos \theta, y=\frac{1}{3}\left(\gamma_{1}-\gamma_{2}\right) \sin \theta
$$

In the scenario
$$
\begin{array}{l} 
x^{2} \sin ^{2} \theta-y^{2} \cos ^{2} \theta=\frac{4}{9} \gamma_{1} \gamma_{2} \sin \theta \\
-\cos \theta, S=\frac{1}{2} \gamma_{1} \gamma_{2} \sin 2 \theta .
\end{array}
$$

After rearranging, we get
$$
\begin{array}{l}
\frac{x^{2}}{\left(\frac{2}{3} \operatorname{ctg} \theta\right)^{2} S}-\frac{y^{2}}{\left(\frac{2}{3} \operatorname{tg} \theta\right)^{2} S}=1 . \\
\left(0<\theta<\frac{\pi}{2}\right)
\end{array}
$$
The trajectory of $Z$ is the branch of this hyperbola, and the distance from the vertex to the origin, $\frac{2}{2} \operatorname{ctg} \theta \sqrt{S}$, is the minimum value of $|Z|$.