Let \({}^{n}C_{r-1}=28,{}^{n}C_{r}=56\text{ and }{}^{n}C_{r+1}=70\). Let \(\mathrm{A}(4\cos t,4\sin t),\mathrm{B}(2\math…

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

[JEE Main 2025, 28 Jan (Shift 1)]