JEE Main 2021 · 25 July · evening · Numerical answer · +4 / −0
If the lines (x - k)/1 = (y - 2)/2 = (z - 3)/3 and
(x + 1)/3 = (y + 2)/2 = (z + 3)/1 are co-planar, then the value of k is _____________.
Answer and solution
Answer 1
Solution
Maths sir · 4 steps
Two lines are coplanar exactly when the gap between them has no component along their common normal.
- Read off the lines. The first line passes through A(k, 2, 3) with direction d₁ = (1, 2, 3). The second passes through B(-1, -2, -3) with direction d₂ = (3, 2, 1).
- Coplanarity condition. The two directions are not parallel, so the lines are coplanar only if AB⃗, d₁ and d₂ lie in one plane. That means their scalar triple product is zero. Here AB⃗ = (-1-k, -4, -6).
|-1-k, -4, -6; 1, 2, 3; 3, 2, 1| = 0
- Expand. Along the first row: (-1-k)(2-6) + 4(1-9) - 6(2-6) = 4 + 4k - 32 + 24 = 4k - 4.
- So. 4k - 4 = 0 gives k = 1.
The common slip: -1. Reading the point on the first line as (-k, 2, 3) instead of (k, 2, 3) flips a sign and gives k = -1.
The idea to keep. For two non-parallel lines, write the vector joining one point to the other and set the triple product with both directions to zero. Read the point from x - a as +a.
Answer from the NTA final answer key. Solution by PrepWiser.