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JEE MainMathematicsThree Dimensional Geometry

Let the line L₁ be given by x-2 1 = y-1 -1 = z+1 2 and the line L₂ be given by x-4 2 = y-3 1 = z-1 -1 . A line L is parallel to the vector i + j + k and intersects L₁ and L₂ at the points C and D , respectively. If a plane passes through the origin O(0,0,0) and the points C and D , and the perpendicular distance of the point P₀(5, -2, 4) from is d , then 14d^2 is equal to :

Options

  1. A400
  2. B64
  3. C144
  4. D0

Correct answer

A. 400

Step-by-step solution

Let the coordinates of C on L₁ be ( +2, - +1, 2 -1) and D on L₂ be (2 +4, +3, - +1) . The direction ratios of the line CD are proportional to (1, 1, 1) . CD = (2 - + 2) i + ( + + 2) j + (- - 2 + 2) k Thus, 2 - + 2 = + + 2 = - - 2 + 2 . From the first two parts: 2 - = + = 2 . From the last two parts: + = - - 2 2 + 3 = 0 . Substituting = 2 gives 4 + 3 = 0 = 0 , which means = 0 . So, C is (2, 1, -1) and D is (4, 3, 1) . The plane passes through O(0,0,0) , C(2, 1, -1) , and D(4, 3, 1) . A normal vector to is n = OC OD

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