Mathematics Question from JEE Main 2025 · 29 January, Shift 2

Let $y^2 = 12x$ be the parabola and $S$ its focus. Let $PQ$ be a focal chord of the parabola such that $(SP)(SQ) = \frac{147}{4}$.

Let $C$ be the circle described taking $PQ$ as a diameter. If the equation of a circle $C$ is

$$64x^2 + 64y^2 - \alpha x - 64\sqrt{3}y = \beta$$

then $\beta - \alpha$ is equal to: