Let \( H : \frac{-x^2}{a^2} + \frac{y^2}{b^2} = 1 \) be the hyperbola, whose eccentricity is \( \sqrt{3} \) and the length of the latus rectum is \( 4\sqrt{3} \). Suppose the point \( (\alpha, 6) \), \( \alpha > 0 \) lies on \( H \). If \( \beta \) is the product of the focal distances of the point \( (\alpha, 6) \), then \( \alpha^2 + \beta \) is equal to:
