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