JEE Main 2023 · 13 April · evening · Single correct · +4 / −1
The line, that is coplanar to the line (x+3)/-3=(y-1)/1=(z-5)/5, is :
- (a) (x+1)/-1=(y-2)/2=(z-5)/4
- (b) (x+1)/-1=(y-2)/2=(z-5)/5
- (c) (x-1)/-1=(y-2)/2=(z-5)/5
- (d) (x+1)/1=(y-2)/2=(z-5)/5
Answer and solution
Answer (b) (x+1)/-1=(y-2)/2=(z-5)/5
Solution
Maths sir · 3 steps
Two lines are coplanar exactly when the scalar triple product of their joining vector and directions is zero.
- Coplanarity test. The given line passes through A(-3, 1, 5) with direction d⃗₁ = (-3, 1, 5). A second line through B with direction d⃗₂ lies in the same plane exactly when AB⃗ · (d⃗₁ × d⃗₂) = 0.
|x₂ - x₁, y₂ - y₁, z₂ - z₁; -3, 1, 5; a₂, b₂, c₂| = 0
Option AB⃗ d⃗₁ × d⃗₂ Triple product (a) (2, 1, 0) (-6, 7, -5) -12 + 7 = -5 (b) (2, 1, 0) (-5, 10, -5) -10 + 10 = 0 (c) (4, 1, 0) (-5, 10, -5) -20 + 10 = -10 (d) (2, 1, 0) (-5, 20, -7) -10 + 20 = 10 - Quick check. In both the given line and (b), the point and the direction have the same numbers. So putting the parameter equal to -1 gives the origin in both lines. The two lines meet at (0, 0, 0), and lines that meet are always coplanar.
- So. Only option (b) makes the triple product zero, so it is the line coplanar with the given one.
The idea to keep. For coplanar lines, put the joining vector and both directions in a 3 by 3 determinant and check that it is zero. If both lines pass through a common point, they are coplanar at once.
Answer from the NTA final answer key. Solution by PrepWiser.