JEE 2023 — Mathematics PYQ

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Question

JEE 2023 — Mathematics PYQ

JEE | Mathematics | 2023

A plane P contains the line of intersection of the plane $r \cdot (\hat{i} + \hat{j} + \hat{k}) = 6$ and $r \cdot (2 \hat{i} + 3 \hat{j} + 4 \hat{k}) = - 5$ . If P passes through the point $(0, 2, - 2)$ , then the square of distance of the point $(12, 12, 18)$ from the plane P is:

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Explanation

Given plane $r (\hat{i} + \hat{j} + \hat{k}) = 6$ and
$r (2 \hat{i} + 3 \hat{j} + 4 \hat{k}) = - 5$

Equation of plane passing through both planes
P1 $\to (x \hat{i} + y \hat{j} + z \hat{k}) (\hat{i} + \hat{j} + \hat{k}) = 6$
P1 $= x + y + z = 6$
P2 $\to (x \hat{i} + y \hat{j} + z \hat{k}) (2 \hat{i} + 3 \hat{j} + 4 \hat{k}) = - 5$
P2 $= 2 x + 3 y + 4 z = - 5$
P1 + λP2 = 0
$(x+y+z-6)+λ(2x+3y+4z+5)=0$
passes through (0,2,-2)
$\Rightarrow(0+2-2-6)+λ(2×0+3×2+4×(-2)+5)=0$
$\Rightarrowλ=2$
Equation of plane
$5 x + 7 y + 9 z + 4 = 0$
Distance (12,12,18) is
$d=\frac{|5×12+7×12+9×18+4|}{\sqrt{5^2+7^2+9^2}}$
$d = \frac{310}{155}$
$\Rightarrow d^{2} = 620$

Choose the correct answer:

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