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Two lines L₁ and L₂ pass through the point A(1, 1, 1) , and their direction cosines satisfy the equations 2l - m + 2n = 0 and mn + nl + lm = 0 . If Q₁ and Q₂ are the images of the point P(1, -2, 4) in the lines L₁ and L₂ respectively, then the value of (PQ₁)^2 + (PQ₂)^2 is equal to:

Options

  1. A27
  2. B54
  3. C72
  4. D108

Correct answer

D. 108

Step-by-step solution

The direction cosines of the lines satisfy the given equations: 2l - m + 2n = 0 m = 2l + 2n Substitute m into the second equation mn + nl + lm = 0 : (2l + 2n)n + nl + l(2l + 2n) = 0 2ln + 2n^2 + nl + 2l^2 + 2ln = 0 2l^2 + 5ln + 2n^2 = 0 (2l + n)(l + 2n) = 0 Case 1: n = -2l Then m = 2l + 2(-2l) = -2l . The direction ratios of L₁ are (l, -2l, -2l) , which simplifies to (1, -2, -2) . The unit direction vector is u ₁ = 1 3 i - 2 3 j - 2 3 k . Case 2: l = -2n Then m = 2(-2n) + 2n = -2n . The direction ratios of L₂ are (

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