Consider the lines L 1 and L 2 given by L 1 : x - 1 2 = y - 3 1 = z - 2 2 , L 2 : x - 2 1 = y - 2 2 = z - 3 3 . A line L 3 having direction ratios 1 , - 1 , - 2 intersects L 1 and L 2 at the points P and Q respectively. Then the length o

STUDY MATERIALS
COURSES
MORE
SUCCESS STORY
ADMISSION
STORE

Back

JEE ADVANCE

CLASS XI - XII
CRASH COURSE

CLASS XI - XII

CRASH COURSE

JEE MAIN

CLASS XI - XII
CRASH COURSE

CLASS XI - XII

CRASH COURSE

NEET

CLASS XI - XII
CRASH COURSE

CLASS XI - XII

CRASH COURSE

BITSAT

CLASS XI - XII

CLASS XI - XII

CBSE BOARD

CLASS IX

CLASS X

CLASS XI

CLASS XII

CLASS VI

CLASS VII

CLASS VIII

UP BOARD

CLASS IX
CLASS X
CLASS XI
CLASS XII

CLASS IX

CLASS X

CLASS XI

CLASS XII

CUET

CRASH COURSE

CRASH COURSE

Courses

CLASSROOM COURSES
ONLINE COURSES

ONLINE COURSES

More

Solution

Q.
Consider the lines $L_{1}$ and $L_{2}$ given by $L_{1} : \frac{x - 1}{2} = \frac{y - 3}{1} = \frac{z - 2}{2}, L_{2} : \frac{x - 2}{1} = \frac{y - 2}{2} = \frac{z - 3}{3} .$ A line $L_{3}$ having direction ratios $1, - 1, - 2$ intersects $L_{1}$ and $L_{2}$ at the points P and Q respectively. Then the length of line segment PQ is [2023]

1 $3 \sqrt{2}$

2 $4$

3 $2 \sqrt{6}$

4 $4 \sqrt{3}$

Ans.
(3)

$P lies on \frac{x - 1}{2} = \frac{y - 3}{1} = \frac{z - 2}{2} = λ$

$P \equiv (2 λ + 1, λ + 3, 2 λ + 2)$

$Q lies on \frac{x - 2}{1} = \frac{y - 2}{2} = \frac{z - 3}{3} = μ$

$Q \equiv (μ + 2, 2 μ + 2, 3 μ + 3)$

$D.R.'s of P Q = (2 λ - μ - 1, λ - 2 μ + 1, 2 λ - 3 μ - 1)$

$which are proportional to 1, - 1, - 2$

$\frac{2 λ - μ - 1}{1} = \frac{λ - 2 μ + 1}{- 1} = \frac{2 λ - 3 μ - 1}{- 2}$

$- 2 λ + μ + 1 = λ - 2 μ + 1 \Rightarrow λ = μ$

$- 2 λ + 4 μ - 2 = - 2 λ + 3 μ + 1 \Rightarrow μ = 3 = λ$

$P = (7, 6, 8), Q = (5, 8, 12)$

$P Q^{2} = 2^{2} + 2^{2} + 4^{2} = 24 \Rightarrow P Q = 2 \sqrt{6}$

Want Us to Email you About Special Offers & Updates?