Let \(A=\left[\begin{matrix}1 & 2 & 7 \\ 4 & -2 & 8 \\ 3 & 8 & -7\end{matrix}\right]\) and \(\det (\mathrm{A}-\alpha \ma…

Let \(A=\left[\begin{matrix}1 & 2 & 7 \\ 4 & -2 & 8 \\ 3 & 8 & -7\end{matrix}\right]\) and \(\det (\mathrm{A}-\alpha \mathrm{I})=0\), where \(\alpha\) is a real number. If the largest possible value of \(\alpha\) is \(p\), then the circle \((x-p{)}^{2}+(y-2p{)}^{2}=320\), intersects the co-ordinate axes at

[JEE Main 2026, 4 Apr (Shift 2)]