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Let Q be the mirror image of the point P(6, 2, 2) in the line L₁: x-1 1 = y-1 2 = z-1 2 . Let L₂ be a line passing through Q and the origin (0,0,0) . The shortest distance between the line L₂ and the line L₃: x 2 = y-6 2 = z -1 is:

Options

  1. A18
  2. B3 2
  3. C2
  4. D4 3

Correct answer

C. 2

Step-by-step solution

Let F be the foot of the perpendicular from P(6, 2, 2) to the line L₁ . A general point on L₁ can be written as F( +1, 2 +1, 2 +1) . The direction ratios of PF are ( -5, 2 -1, 2 -1) . Since PF is perpendicular to L₁ , its dot product with the direction vector of L₁ , which is (1, 2, 2) , must be zero: 1( -5) + 2(2 -1) + 2(2 -1) = 0 - 5 + 4 - 2 + 4 - 2 = 0 9 - 9 = 0 = 1 . Thus, the coordinates of F are (2, 3, 3) . Since F is the midpoint of P and its image Q , we have Q = 2F - P = (4, 6, 6) - (6, 2, 2) = (-2, 4, 4)

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