Conic Sections · Hyperbola foci and directrices

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

\dfrac83

. If the line

x = \alpha

intersects the hyperbola

H

at the points

A

and

B

such that the area of the triangle

AOB

4\sqrt{15}

, where

O

is the origin, then

\alpha^2

equals

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