Mathematics Question from JEE Main 2024 · 30 January, Shift 2

Consider two circles $C_1$: $x^2 + y^2 = 25$ and $C_2$: $(x-\alpha)^2 + y^2 = 16$, where $\alpha \in (5, 9)$. Let the angle between the two radii (one to each circle) drawn from one of the intersection points of $C_1$ and $C_2$ be $\sin^{-1}\left(\frac{\sqrt{63}}{8}\right)$. If the length of the common chord of $C_1$ and $C_2$ is $\beta$, then the value of $(\alpha\beta)^2$ equals: