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