Let ell 1 and ell 2 be the lines vec r 1 =λ(hat i +hat j +hat k ) and overrightarrow r 2 =(hat j -hat k )+μ(hat i +hat k ) , respectively. Let X be the set of all the planes H that contain the line ell 1 . For a plane H , let d ( H ) denote the smallest possible distance between the points of ell 2 and H . Let H 0 be a plane in X for which d ( H 0 ) is the maximum value of d ( H ) as H varies over all planes in X .Match each entry in List-I to the correct entries in List-II. LIST-I LIST-II --- --- (P) The value of d ( H 0 ) is 1) √(3) (Q) The distance of the point (0,1,2) from H 0 is 2) frac 1 √(3) (R) The distance of origin from H 0 is 3) 0 (S) The distance of origin from the point of intersection of planes y=z, x=1 and H 0 is 4) √(2) 5) frac 1 √(2)

( P ) rightarrow(5) ( Q ) rightarrow(4) ( R ) rightarrow(3) (S) rightarrow (1)

( P ) rightarrow(2) ( Q ) rightarrow(4) ( R ) rightarrow(5) (S) rightarrow (1)

( P ) rightarrow(2) ( Q ) rightarrow(1) ( R ) rightarrow(3) (S) rightarrow (2)

( P ) rightarrow(5) ( Q ) rightarrow(1) ( R ) rightarrow(4) (S) rightarrow (2)

Mathematics · 3D Geometry

JEE Advanced 2023 — Paper 1 — Question 15

Let $\ell_{1}$ and $\ell_{2}$ be the lines $\vec{r}_{1}=\lambda(\hat{\mathrm{i}}+\hat{\mathrm{j}}+\hat{\mathrm{k}})$ and $\overrightarrow{\mathrm{r}}_{2}=(\hat{\mathrm{j}}-\hat{\mathrm{k}})+\mu(\hat{\mathrm{i}}+\hat{\mathrm{k}})$ , respectively. Let X be the set of all the planes H that contain the line $\ell_{1}$ . For a plane H , let $\mathrm{d}(\mathrm{H})$ denote the smallest possible distance between the points of $\ell_{2}$ and H . Let $\mathrm{H}_{0}$ be a plane in X for which $\mathrm{d}\left(\mathrm{H}_{0}\right)$ is the maximum value of $\mathrm{d}(\mathrm{H})$ as H varies over all planes in X .Match each entry in List-I to the correct entries in List-II.

LIST-I	LIST-II
(P) The value of $\mathrm{d}\left(\mathrm{H}_{0}\right)$ is	1) $\sqrt{3}$
(Q) The distance of the point $(0,1,2)$ from $\mathrm{H}_{0}$ is	2) $\frac{1}{\sqrt{3}}$
(R) The distance of origin from $\mathrm{H}_{0}$ is	3) 0
(S) The distance of origin from the point of intersection of planes $y=z, x=1$ and $H_{0}$ is	4) $\sqrt{2}$
	5) $\frac{1}{\sqrt{2}}$

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

Answer: B

Step-by-step solution

Line $l_{2}$ is parallel to plane containing $l_{1}$

Let $l, \mathrm{~m}, \mathrm{n}$ be D . cosine of plane H

\mathrm{l}+\mathrm{m}+\mathrm{n}=0 …(1)

l+n=0 …(2)

- \mathrm{n}+\mathrm{m}+\mathrm{n}=0, \mathrm{~m}=0

$\frac{\mathrm{l}}{\sqrt{2}}=\frac{\mathrm{n}}{-\sqrt{2}}=\frac{\mathrm{m}}{0}$

$\frac{\mathrm{l}}{\sqrt{2}}=\frac{\mathrm{m}}{0}=\frac{\mathrm{n}}{-\sqrt{2}}$

so equation of plane $H_{0}$ is $\sqrt{2}(\mathrm{x}-0)+0(\mathrm{y}-0)-\sqrt{2}(\mathrm{z}-0)=0$

$\mathrm{x}-\mathrm{z}=0$ equation of plane H

(P) $\mathrm{d}\left[\mathrm{H}_{0}\right]=\mathrm{PM}=\frac{0+1}{\sqrt{1+1}}=\frac{1}{\sqrt{2}}$

(Q) $=\left|\frac{0-2}{\sqrt{2}}\right|=\sqrt{2}$

(R) Distance of origin from $\mathrm{H}_{0}=0$

(S) Distance of origin from the point of intersection of planes $y=z, x=1, x=z$ . Intersection $T(1,1,1)$ ,

Distance from origin $=\sqrt{1+1+1}=\sqrt{3}$

Answer key and solution verified before publishing.

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Exam: JEE Advanced 2023
Paper: Paper 1
Subject: Mathematics
Chapter: 3D Geometry
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Step-by-step solution

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