Mathematics Question from JEE Main 2025 · 28 January, Shift 1

Let ${}^nC_{r-1} = 28$, ${}^nC_r = 56$ and ${}^nC_{r+1} = 70$. Let A $\left(4\cos t, 4\sin t\right)$, B $\left(2\sin t, -2\cos t\right)$ and C $\left(3r - n,\ r^2 - n - 1\right)$ be the vertices of a triangle ABC, where $t$ is a parameter. If $\left(3x - 1\right)^2 + \left(3y\right)^2 = \alpha$ is the locus of the centroid of triangle ABC, then $\alpha$ equals: