Complex Numbers and Quadratic Equations: JEE Main 2025 Mathematics question

Let $\left| \frac{z - \overline{z}}{2z + \overline{z}} \right| = \frac{1}{3}$, $z \in \mathbb{C}$, be the equation of a circle with center at $C$. If the area of the triangle, whose vertices are at the points $(0, 0)$, $C$ and $(\alpha, 0)$ is 11 square units, then $\alpha^2$ equals