Mathematics · 3D Geometry

JEE Advanced 2024 — Paper 1 — Question 16

Let $\gamma \in R$ be such that the lines $L_{1}: \frac{x+11}{1}=\frac{y+21}{2}=\frac{z+29}{3}$ and $L_{2}: \frac{x+16}{3}=\frac{y+11}{2}=\frac{z+4}{\gamma}$ intersect. Let $R_{1}$ be the point of intersection of $L_{1}$ and $L_{2}$ . Let $O=(0,0,0)$ , and $\hat{n}$ denote a unit normal vector to the plane containing both the lines $L_{1}$ and $L_{2}$ .

Match each entry in List-I to the correct entries in List-II.

List - I	List - II
(P) $\gamma$ equals	(1) $-\hat{i}-\hat{j}+\hat{k}$
(Q) A possible choice for $\hat{n}$ is	(2) $\sqrt{\frac{3}{2}}$
(R) $\overrightarrow{O R}_{1}$ equals	(3) 1
(S) A possible value of $\overrightarrow{\mathrm{OR}_{1}} \cdot \hat{n}$ is	(4) $\frac{1}{\sqrt{6}} \hat{i}-\frac{2}{\sqrt{6}} \hat{j}+\frac{1}{\sqrt{6}} \hat{k}$
	(5) $\sqrt{\frac{2}{3}}$

The correct option is:

Option A:
(P) $\rightarrow$ (3)(Q) $\rightarrow$ (4) $(R) \rightarrow(1)$ $(S) \rightarrow(2)$
Option B:
$(\mathrm{P}) \rightarrow(5)$$(\mathrm{Q}) \rightarrow(4) \quad(\mathrm{R}) \rightarrow(1)$$(S) \rightarrow(2)$
Option C:
$(\mathrm{P}) \rightarrow(3)$$(\mathrm{Q}) \rightarrow(4)$$(R) \rightarrow(1)$ (S) $\rightarrow$ (5)
Correct
Option D:
$(\mathrm{P}) \rightarrow(3)$$(Q) \rightarrow(1)$$(R) \rightarrow(4)$$(S) \rightarrow(5)$

Answer: C

Step-by-step solution

$\frac{x+11}{1}=\frac{y+21}{2}=\frac{z+29}{3}=\alpha$ $\frac{x+16}{3}=\frac{y+11}{2}=\frac{z+4}{4}=\beta$ $\Rightarrow \alpha-11=3 \beta-16 \Rightarrow \alpha-3 \beta=-5$ $2 \alpha-21=2 \beta-11 \Rightarrow \alpha-\beta=5$ $\alpha=10, \beta=5$ So, $\gamma=1$ $\hat{n}=\left|\begin{array}{lll}\hat{i} & \hat{j} & k\\ 1 & 2 & 3\\ 3 & 2 & 1\end{array}\right|=-4 \hat{i}+8 \hat{j}-4 \hat{k}$ $\hat{n}=\left(\frac{1}{\sqrt{6}} i-\frac{2}{\sqrt{6}} \hat{j}+\frac{1}{\sqrt{6}} \hat{k}\right)$ $\mathrm{R}_{1}(-1,-1,1)$ $\overrightarrow{\mathrm{OR}}=(-\hat{\mathrm{i}}-\hat{\mathrm{j}}+\hat{\mathrm{k}})$ $\overrightarrow{\mathrm{OR}} \cdot \hat{\mathrm{n}}=\sqrt{\frac{2}{3}}$

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Exam: JEE Advanced 2024
Paper: Paper 1
Subject: Mathematics
Chapter: 3D Geometry
Topic: Straight Lines in 3D Geometry

Let R^3 denote the three dimensional space. Take two points P=(1,2,3) and Q=(4,2,7) . Let dist(X, Y) denote the distance between two points…

Answer: C

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