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JEE Main 202623 JanuaryMorning Shift
Exam
JEE Main
Year
2026
Date
23 January
Shift
Morning Shift
Subject
Mathematics
Chapter
Three Dimensional Geometry

JEE Main 2026 · 23 January · Morning Shift · Mathematics · Three Dimensional Geometry

Let the direction cosines of two lines satisfy the equations : 4...

Let the direction cosines of two lines satisfy the equations : 4l+m−n=04 l+m-n=0 and 2mn+10nl+3lm=02 m n+10 n l+3 l m=0. Then the cosine of the acute angle between these lines is :

Options

  1. A20338\frac{20}{3 \sqrt{38}}
  2. B10338\frac{10}{3 \sqrt{38}}
  3. C10738\frac{10}{7 \sqrt{38}}
  4. D1038\frac{10}{\sqrt{38}}

Correct answer

B. 10338\frac{10}{3 \sqrt{38}}

Step-by-step solution

From n=4l+mn = 4l + m, substitute in 2mn+10nl+3lm=02mn + 10nl + 3lm = 0:
40l2+21lm+2m2=040l^2 + 21lm + 2m^2 = 0. Let t=l/mt = l/m: 40t2+21t+2=040t^2 + 21t + 2 = 0.
t=−21±1180t = \frac{-21 \pm 11}{80}, giving t=−1/8t = -1/8 or t=−2/5t = -2/5.
l/m=−1/8l/m = -1/8: direction (l,m,n)=(−1,8,4)(l,m,n) = (-1, 8, 4).
l/m=−2/5l/m = -2/5: direction (l,m,n)=(−2,5,−3)(l,m,n) = (-2, 5, -3).
cos⁡θ=∣2+40−12∣8138=30938=10338\cos\theta = \frac{|2+40-12|}{\sqrt{81}\sqrt{38}} = \frac{30}{9\sqrt{38}} = \frac{10}{3\sqrt{38}}.
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