JEE 2023 — Mathematics PYQ

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Question

JEE 2023 — Mathematics PYQ

JEE | Mathematics | 2023

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 $B (5, α, β), α, β \in Z$ is a point on plane $P$ such that the area of the triangle $A B N$ is $32$ , then $α^{2} + β^{2} + α β$ is equal to ____.

Choose the correct answer:

Explanation

Solution:

Foot of Perpendicular ( $N$ ):

$\frac{x-4}{1} = \frac{y-3}{-1} = \frac{z-1}{2} = -\frac{(4-3+2+3)}{1+1+4} = -1 \implies N(3, 4, -1)$
Distance $AN$ (Height):

$AN = \sqrt{(4-3)^2 + (3-4)^2 + (1+1)^2} = \sqrt{6}$
Area of $\Delta ABN$ :

$\frac{1}{2} \times NB \times \sqrt{6} = 3\sqrt{2} \implies NB = \frac{6\sqrt{2}}{\sqrt{6}} = 2\sqrt{3}$
Point $B$ on Plane:

$5 - \alpha + 2\beta + 3 = 0 \implies \alpha - 2\beta = 8 \dots (i)$

$NB^2 = (5-3)^2 + (\alpha-4)^2 + (\beta+1)^2 = 12 \dots (ii)$

Solving $(i)$ and $(ii)$ with integer values: $\alpha = 2, \beta = -3$ .
Final Value:

$\alpha^2 + \beta^2 + \alpha\beta = (2)^2 + (-3)^2 + (2)(-3) = 4 + 9 - 6 = 7$

Final Answer: 7