Unclassified (maths): JEE Main 2025 Mathematics question

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 $a^2 b^2 = \alpha \sqrt{2} - \beta$, $\alpha, \beta \in \mathbb{N}$, then