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JEE Main 20245 AprilMorning Shift
Exam
JEE Main
Year
2024
Date
5 April
Shift
Morning Shift
Subject
Mathematics
Chapter
Three Dimensional Geometry

JEE Main 2024 · 5 April · Morning Shift · Mathematics · Three Dimensional Geometry

If the line (2-x)/(3)=(3 y-2)/(4 λ+1)=4-z makes a right angle with...

If the line 2−x3=3y−24λ+1=4−z\frac{2-x}{3}=\frac{3 y-2}{4 \lambda+1}=4-z makes a right angle with the line x+33μ=1−2y6=5−z7\frac{x+3}{3 \mu}=\frac{1-2 y}{6}=\frac{5-z}{7}, then 4λ+9μ4 \lambda+9 \mu is equal to :

Options

  1. A4
  2. B13
  3. C5
  4. D6

Correct answer

D. 6

Step-by-step solution

2−x3=3y−24λ+1=4−z...(1)x−2(−3)=y−23(4λ+13)=z−4(−1)\frac{2-x}{3}=\frac{3 y-2}{4 \lambda+1}=4-z...(1) \frac{x-2}{(-3)}=\frac{y-\frac{2}{3}}{\left(\frac{4 \lambda+1}{3}\right)}=\frac{z-4}{(-1)}x+33μ=1−2y6=5−z7...(2)x+33μ=y−12(−3)=z−5(−7) Right angle ⇒(−3)(3μ)+(4λ+13)(−3)+(−1)(−7)=0−9μ−4λ−1+7=04λ+9μ=6\begin{aligned} & \frac{x+3}{3 \mu}=\frac{1-2 y}{6}=\frac{5-z}{7}...(2) \\ & \frac{x+3}{3 \mu}=\frac{y-\frac{1}{2}}{(-3)}=\frac{z-5}{(-7)} \\ & \text{ Right angle } \Rightarrow(-3)(3 \mu)+\left(\frac{4 \lambda+1}{3}\right)(-3)+(-1)(-7)=0 \\ & -9 \mu-4 \lambda-1+7=0 \\ & 4 \lambda+9 \mu=6\end{aligned}$$
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