Let H: x^2/a^2-y^2/b^2=1 be a hyperbola such that the distance between its foci is 6 and the distance between its directrices is 8/3. If the line x=α…

Let $\displaystyle \mathrm{H}: \frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ be a hyperbola such that the distance between its foci is $\displaystyle 6$ and the distance between its directrices is $\displaystyle \frac{8}{3}$. If the line $\displaystyle x=\alpha$ intersects the hyperbola H at the points A and B such that the area of the triangle AOB is $\displaystyle 4 \sqrt{15}$, where O is the origin, then $\displaystyle \alpha^2$ equals