Standard equation of hyperbola and mixed ellipse questionsMedium

Hyperbola Foci Directrix and Area

Let $H: \;\dfrac{x^{2}}{a^{2}}-\;\dfrac{y^{2}}{b^{2}}=1$ be a hyperbola such that the distance between its foci is 6 and the distance between its directrices is $\;\dfrac{8}{3}$ . 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

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