JEE 2023 — Mathematics PYQ

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Question

JEE 2023 — Mathematics PYQ

JEE | Mathematics | 2023

Consider the lines $L_1$ and $L_2$ given by

$L_{1} : \frac{x - 1}{2} = \frac{y - 3}{1} = \frac{z - 2}{2}$

$L_{2} : \frac{x - 2}{1} = \frac{y - 2}{2} = \frac{z - 3}{3}$

L_2: \frac{x-2}{1} = \frac{y-2}{2} = \frac{z-3}{3}

Choose the correct answer:

Explanation

Solution

Points $P$ and $Q$ :

$P = (2\lambda + 1, \lambda + 3, 2\lambda + 2)$

$Q = (\mu + 2, 2\mu + 2, 3\mu + 3)$

Direction Ratios of $PQ$ :

$\vec{PQ} = ((\mu + 2) - (2\lambda + 1), (2\mu + 2) - (\lambda + 3), (3\mu + 3) - (2\lambda + 2))$

$\vec{PQ} = (\mu - 2\lambda + 1, 2\mu - \lambda - 1, 3\mu - 2\lambda + 1)$

Comparing with D.R.s of $L_3$ $(1, -1, -2)$ :

$\frac{\mu - 2\lambda + 1}{1} = \frac{2\mu - \lambda - 1}{-1} = \frac{3\mu - 2\lambda + 1}{-2} = k$

From first two:

$-(\mu - 2\lambda + 1) = 2\mu - \lambda - 1$

$-\mu + 2\lambda - 1 = 2\mu - \lambda - 1$

$3\lambda = 3\mu \implies \lambda = \mu$

Using $\lambda = \mu$ in first and third:

$\frac{\mu - 2\mu + 1}{1} = \frac{3\mu - 2\mu + 1}{-2}$

$-2(-\mu + 1) = \mu + 1$

$2\mu - 2 = \mu + 1 \implies \mu = 3$

$\therefore \lambda = 3, \mu = 3$

Coordinates:

$P = (7, 6, 8)$

$Q = (5, 8, 12)$

Length $PQ$ :

$PQ = \sqrt{(5-7)^2 + (8-6)^2 + (12-8)^2}$

$PQ = \sqrt{(-2)^2 + (2)^2 + (4)^2}$

$PQ = \sqrt{4 + 4 + 16} = \sqrt{24}$

$PQ = 2\sqrt{6}$

Correct Option: (4)

Choose the correct answer:

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