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

A

172

B

171

C

169

D

170