Mathematics Question from JEE Main 2023 · 10 April, Shift 2

Let the foot of perpendicular from the point $A(4, 3, 1)$ on the plane $P : x - y + 2z + 3 = 0$ be $N$. If $B(5, \alpha, \beta)$ is a point on plane $P$ such that the area of triangle $ABN$ is $3\sqrt{2}$, then $\alpha^2 + \beta^2 + \alpha \beta$ is equal to: