14. In the complex plane, $O$ is the origin, and two moving points $Z_{1}, Z_{2}$ satisfy: (1) The arguments of the complex numbers corresponding to $Z_{1}$ and $Z_{2}$ are fixed values $\theta$ and $-\theta\left(0<\theta<\frac{\pi}{2}\right)$; (2) The area of $\triangle O Z_{1} Z_{2}$ is a fixed value $S$. Find the locus of the centroid $Z$ of $\triangle O Z_{1} Z_{2}$.
14. Let $z_{1}=r_{1}(\cos \theta+\mathrm{i} \sin \theta)$
$$
\begin{aligned}
z_{2} & =r_{2}(\cos \theta-i \sin \theta) \\
z & =\frac{r_{1}+r_{2}}{3} \cos \theta+\frac{r_{1}-r_{2}}{3} i \sin \theta
\end{aligned}
$$

By the problem, $r_{1} r_{2}$ is a constant.
$$
\text { Let } \frac{r_{1}+r_{2}}{3}=x, \frac{r_{1}-r_{2}}{3}=y \text {, }
$$

So $x^{2}-y^{2}=\frac{4 r_{1} r_{2}}{9}$ is a constant,
Therefore, the required locus is a hyperbola.