JEE Main 2020 · 5 September · evening · Single correct · +4 / −1

If for some α ∈ R, the lines
L₁ : (x + 1)/2 = (y - 2)/(- 1) = (z - 1)/1 and
L₂ : (x + 2)/α = (y + 1)/(5 - α) = (z + 1)/1 are coplanar,
then the line L₂ passes through the point :

  1. (a) (10, 2, 2)
  2. (b) (2, –10, –2)
  3. (c) (10, –2, –2)
  4. (d) (–2, 10, 2)
Answer and solution

Answer (b) (2, –10, –2)

Solution

Maths sir · 4 steps

Coplanar lines make the scalar triple product zero. That one condition fixes α.

  1. Read off the lines. L₁ passes through A(-1, 2, 1) with direction (2, -1, 1). L₂ passes through B(-2, -1, -1) with direction (α, 5-α, 1). So AB⃗ = (-1, -3, -2).
  2. Coplanar condition. Two lines are coplanar when AB⃗ and the two direction vectors lie in one plane. That happens exactly when their determinant is zero.

    |-1, -3, -2; 2, -1, 1; α, 5-α, 1| = -(α - 6) + 3(2-α) - 2(10-α) = -2α - 8 = 0

  3. Find the line. So α = -4, and L₂ becomes (x+2)/-4 = (y+1)/9 = (z+1)/1. A point lies on L₂ only if all three ratios come out equal.
    Point(x+2)/-4(y+1)/9(z+1)/1
    (10, 2, 2)-31/33
    (2, -10, -2)-1-1-1
    (10, -2, -2)-3-1/9-1
    (-2, 10, 2)011/93
  4. So. Only (2, -10, -2) gives the same value -1 for all three ratios, so L₂ passes through it. The answer is (b).

The idea to keep. For coplanar lines, set the determinant of (joining vector, direction 1, direction 2) to zero. To test a point on a line, all three ratios must be equal.

Answer from the NTA final answer key. Solution by PrepWiser.

Study Three Dimensional Geometry

Asked about 300 times in JEE Main since 2020. This question is from 2020.