JEE 2023 — Mathematics PYQ

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Question

JEE 2023 — Mathematics PYQ

JEE | Mathematics | 2023

Find the shortest distance between line $l_1$ (passing through $(2,6,2)$ and perpendicular to $2x + y - 2z = 10$ ) and line $l_2: \frac{x+1}{2} = \frac{y+4}{-3} = \frac{z}{2}$ .

Choose the correct answer:

Explanation

Solution:

Equation of line $l_1$ :

Since it's perpendicular to the plane, its direction ratios are $(2, 1, -2)$ .

$l_1: \frac{x-2}{2} = \frac{y-6}{1} = \frac{z-2}{-2}$
Shortest Distance (SD) Formula:

Line 1: $\vec{a}_1 = (2, 6, 2), \vec{b}_1 = (2, 1, -2)$

Line 2: $\vec{a}_2 = (-1, -4, 0), \vec{b}_2 = (2, -3, 2)$

$\vec{a}_2 - \vec{a}_1 = (-3, -10, -2)$

$\vec{b}_1 \times \vec{b}_2 = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 2 & 1 & -2 \\ 2 & -3 & 2 \end{vmatrix} = (-4, -8, -8)$
Calculation:

$|\vec{b}_1 \times \vec{b}_2| = \sqrt{(-4)^2 + (-8)^2 + (-8)^2} = \sqrt{16+64+64} = 12$

$SD = \frac{|(\vec{a}_2 - \vec{a}_1) \cdot (\vec{b}_1 \times \vec{b}_2)|}{|\vec{b}_1 \times \vec{b}_2|} = \frac{|12 + 80 + 16|}{12} = \frac{108}{12} = 9$

Correct Option: (4) 9

Choose the correct answer:

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