Mathematics · 3D Geometry

JEE Advanced 2024 — Paper 2 — Question 6

A straight line drawn from the point $P(1,3,2)$ , parallel to the line $\frac{x-2}{1}=\frac{y-4}{2}=\frac{z-6}{1}$ intersects the plane $L_{1}: x-y+3 z=6$ at the point $Q$ . Another straight line which passesthrough $Q$ and is perpendicular to the plane $L_{1}$ intersects the plane $L_{2}: 2 x-y+z=-4$ at the point $R$ . then which of the following statements is(are) TRUE?

Option A:
The length of the line segment $P Q$ is $\sqrt{6}$
Correct
Option B:
The coordinates of R are $(1,6,3)$
Option C:
The centroid of the triangle PQR is $\left(\frac{4}{3}, \frac{14}{3}, \frac{5}{3}\right)$
Correct
Option D:
The perimeter of the triangle PQR is $\sqrt{2}+\sqrt{6}+\sqrt{11}$

Answer: A, C

Step-by-step solution

Equation of line passing through $P(1,3,2)$ parallel to the line $\frac{x-2}{1}=\frac{y-4}{2}=\frac{z-6}{1}$ is $\frac{x-1}{1}=\frac{y-3}{2}=\frac{z-2}{1}$

$\Rightarrow$ Point $Q$ is $(2,5,3)$ Line passing through $Q(2,5,3)$ and perpendicular to the plane $L_{1}: x-y+3 z=6$ is $\frac{x-2}{1}=\frac{y-5}{-1}=\frac{z-3}{3}$ $\Rightarrow$ Point $R$ is $(1,6,0)$ $\Rightarrow$ Length $P Q$ is $\sqrt{6}$

$\Rightarrow$ Centroid of triangle $P Q R$ is $\left(\frac{4}{3}, \frac{14}{3}, \frac{5}{3}\right)$

$\Rightarrow$ Perimeter of triangle PQR is $\sqrt{6}+\sqrt{11}+\sqrt{13}$

Answer key and solution verified before publishing.

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Exam: JEE Advanced 2024
Paper: Paper 2
Subject: Mathematics
Chapter: 3D Geometry
Topic: Intersection of lines, line & plane.