15. In the complex plane, the three vertices of $\triangle A B C$ correspond to the complex numbers $z_{1}, z_{2}, \dot{z}_{3}$. It is known that $\left|z_{1}\right|=3$, $z_{2}=\bar{z}_{1}, z_{3}=\frac{1}{z_{1}}$, find the maximum value of the area of $\triangle A B C$.
15. Let $z_{1}=3(\cos \theta+i \sin \theta)$, then $z_{2}=3(\cos \theta-i \sin \theta), z_{3}=\frac{1}{3}(\cos \theta-i \sin \theta)$. This indicates that $\overrightarrow{O B}, \overrightarrow{O C}$ are in the same direction. Therefore, $S_{\triangle A B C}=S_{\triangle A O B}-S_{\triangle A C C}=\frac{1}{2}|O A| \cdot|O B| \cdot|\sin 2 \theta|-\frac{1}{2}|O A| \cdot|O C| \cdot|\sin 2 \theta|=4|\sin 2 \theta| \leqslant$ 4 , hence when $\theta=\frac{\pi}{2}$, the maximum value of $S_{\triangle A B C}$ is 4 ,