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Let the line L₁ be given by x-1 1 = y-2 1 = z-3 2 and the line L₂ be given by x-4 1 = y+1 2 = z-6 1 . A line L₃ passes through the point P(2, 1, 4) and intersects L₁ and L₂ at the points C and D , respectively. Then the value of | CD |^2 is equal to :

Options

  1. A3
  2. B12
  3. C9
  4. D27

Correct answer

D. 27

Step-by-step solution

Let the coordinates of C on L₁ be ( +1, +2, 2 +3) and D on L₂ be ( +4, 2 -1, +6) . Since P(2, 1, 4) , C , and D are collinear, the direction ratios of PC and PD are proportional. Direction ratios of PC are ( -1, +1, 2 -1) . Direction ratios of PD are ( +2, 2 -2, +2) . From proportionality, we have: -1 +2 = +1 2 -2 = 2 -1 +2 From the first and third terms: - 1 = 2 - 1 = 0 Substitute = 0 into the first two terms: -1 +2 = 1 2 -2 -2 + 2 = + 2 = 0 Thus, the points are C(1, 2, 3) and D(4, -1, 6) . The vector CD = (4-1) i

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