Let the foot of perpendicular from the point A(4,3,1) on the plane P: x-y+2 z+3=0 be N. If B (5, α, β), α, β Z is a point on plane P such that the area of the triangle ABN is 3 √(2), then α²+β²+α β is equal to _ _ _ _ _ _.

Three Dimensional Geometry: JEE Main 2023 Mathematics question

JEE Main 2023 · 10 Apr, Evening shift · Q22

JEE Main 202310 Apr, Evening shiftMathematicsMediumThree Dimensional Geometry249× since 2002

Let the foot of perpendicular from the point

\mathrm{A}(4,3,1)

on the plane

\mathrm{P}: x-y+2 z+3=0

be N. If B

(5, \alpha, \beta), \alpha, \beta \in \mathbb{Z}

is a point on plane P such that the area of the triangle ABN is

3 \sqrt{2}

, then

\alpha^{2}+\beta^{2}+\alpha \beta

is equal to ___________.

Your answerType your answer. Integers and decimals are accepted.