Consider the hyperbola
$
\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1,
$
having one of its foci at $ P(-3, 0) $. If the latus rectum through its other focus subtends a right angle at $ P $, and
$
a^2b^2 = \alpha\sqrt{2} - \beta, \quad \alpha, \beta \in \mathbb{N},
$
then find $ \alpha $ and $ \beta $.