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JEE MainMathematicsThree Dimensional Geometry

Let the lines L₁ : x-1 2 = y-2 3 = z-3 4 and L₂ : x-8 5 = y-7 2 = z-8 1 intersect at the point P . If the point P' is the image of P in the plane x + y + z = 3 , then the distance of P' from the point Q(-1, 0, 11) is

Options

  1. A13
  2. B65
  3. C57
  4. D329

Correct answer

A. 13

Step-by-step solution

Let the coordinates of point P on L₁ be (2 + 1, 3 + 2, 4 + 3) and on L₂ be (5 + 8, 2 + 7, + 8) . At the point of intersection, we have: 2 + 1 = 5 + 8 2 - 5 = 7 3 + 2 = 2 + 7 3 - 2 = 5 4 + 3 = + 8 4 - = 5 Solving 3 - 2 = 5 and 4 - = 5 , we get = 1 and = -1 . These values also satisfy the first equation. Thus, the intersection point is P(3, 5, 7) . Let P'(x', y', z') be the image of P in the plane x + y + z = 3 . Using the image formula: x' - 3 1 = y' - 5 1 = z' - 7 1 = -2 (3 + 5 + 7 - 3) 1^2 + 1^2 + 1^2 x' - 3 1 = y

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