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Let P be the foot of the perpendicular from the point A(3, -2, 5) on the line L₁ : x-2 2 = y+1 1 = z-1 2 . A line L₂ passes through the points A and P . If the line L₂ intersects the plane : 2x - y + z = 5 at the point Q , then the distance PQ is equal to :

Options

  1. A9
  2. B12
  3. C3
  4. D81

Correct answer

A. 9

Step-by-step solution

Let the general point on L₁ be P(2 +2, -1, 2 +1) . The direction ratios of L₁ are 2, 1, 2 . The direction ratios of AP are 2 -1, +1, 2 -4 . Since AP L₁ , we have : 2(2 -1) + 1( +1) + 2(2 -4) = 0 4 - 2 + + 1 + 4 - 8 = 0 9 - 9 = 0 = 1 Thus, the coordinates of P are (4, 0, 3) . The line L₂ passes through A(3, -2, 5) and P(4, 0, 3) . The direction ratios of L₂ are 4-3, 0-(-2), 3-5 , which are 1, 2, -2 . The equation of L₂ is x-4 1 = y-0 2 = z-3 -2 = t . A general point on L₂ is R(t+4, 2t, -2t+3) . Since Q is the inters

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