JEE Main 2023 · 25 January · morning · Single correct · +4 / −1

Consider the lines L₁ and L₂ given by
L₁:(x - 1)/2 = (y - 3)/1 = (z - 2)/2
L₂:(x - 2)/1 = (y - 2)/2 = (z - 3)/3.
A line L₃ having direction ratios 1, -1, -2, intersects L₁ and L₂ at the points P and Q respectively. Then the length of line segment PQ is

  1. (a) 4√3
  2. (b) 2√6
  3. (c) 4
  4. (d) 3√2
Answer and solution

Answer (b) 2√6

Solution

Maths sir · 5 steps

Write P and Q as general points on their lines, then force PQ to be parallel to (1, −1, −2).

  1. General points. Any point of L₁ is P(1+2λ, 3+λ, 2+2λ) and any point of L₂ is Q(2+μ, 2+2μ, 3+3μ). So PQ⃗ = (1+μ-2λ, -1+2μ-λ, 1+3μ-2λ).
  2. Parallel condition. PQ lies along L₃, so PQ⃗ = k(1, -1, -2) for some number k. That gives three equations in λ, μ and k.

    1+μ-2λ = k, -1+2μ-λ = -k, 1+3μ-2λ = -2k

  3. Solve. Subtracting the first equation from the third gives 2μ = -3k. Putting μ = -3k/2 and λ = 1/2 - 5k/4 into the second gives k = -2, so λ = 3 and μ = 3.
  4. Find the points. P = (7, 6, 8) and Q = (5, 8, 12), so PQ⃗ = (-2, 2, 4), which is -2(1,-1,-2) as required.
  5. So. PQ = √(4 + 4 + 16) = √24 = 2√6, which is option (b).

The idea to keep. When a line meets two given lines, take a general point on each with its own parameter. The direction of the joining vector gives you the equations to find both parameters.

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

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Asked about 300 times in JEE Main since 2020. This question is from 2023.