JEE Main 2021 · 27 August · evening · Single correct · +4 / −1
The angle between the straight lines, whose direction cosines are given by the equations 2l + 2m - n = 0 and mn + nl + lm = 0, is :
- (a) π/2
- (b) π - cos ⁻¹( 4/9 )
- (c) cos ⁻¹( 8/9 )
- (d) π/3
Answer and solution
Answer (a) π/2
Solution
Maths sir · 5 steps
Use the linear equation to remove n, and the quadratic splits neatly into the two lines.
- Eliminate n. From the first equation, n = 2l + 2m. Write the second equation as n(l + m) + lm = 0 and substitute.
2(l+m)² + lm = 0 ⇒ 2l² + 5lm + 2m² = 0 ⇒ (2l + m)(l + 2m) = 0
- First line. Take m = -2l. Then n = 2l - 4l = -2l, so the direction ratios are (1, -2, -2).
- Second line. Take l = -2m. Then n = -4m + 2m = -2m, so the direction ratios are (-2, 1, -2).
- Check the angle. The dot product is (1)(-2) + (-2)(1) + (-2)(-2) = -2 - 2 + 4 = 0. A zero dot product means the lines are perpendicular.
- So. The angle between the lines is π/2, option (a).
The idea to keep. With one linear and one quadratic condition on l, m, n, use the linear one to eliminate a variable. The quadratic then factorises, and each factor gives one line.
Answer from the NTA final answer key. Solution by PrepWiser.