Let $\left| \frac{\bar{z} - i}{2\bar{z} + i} \right| = \frac{1}{3}, z \in C$, 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:
