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JEE MainMathematicsThree Dimensional Geometry

Let the line L intersect the lines x-1 = y-2 = 3-z and x-3 = 4-y = z-4 , and be parallel to the vector 2 i + 2 j + k . If the line L intersects the plane x + y - z = 6 at the point P( , , ) , then the value of + + is

Options

  1. A5
  2. B16
  3. C6
  4. D11

Correct answer

B. 16

Step-by-step solution

Let the given lines be L₁ and L₂ . L₁: x-1 1 = y-2 1 = z-3 -1 = L₂: x-3 1 = y-4 -1 = z-4 1 = A general point on L₁ is A( +1, +2, - +3) and on L₂ is B( +3, - +4, +4) . The direction ratios of the line segment AB are - + 2, - - + 2, + + 1 . Since L is parallel to the vector 2 i + 2 j + k , the direction ratios of AB must be proportional to 2, 2, 1 . - + 2 2 = - - + 2 2 = + + 1 1 From the first two terms: - + 2 = - - + 2 2 = 0 = 0 Substituting = 0 into the first and third terms: - + 2 2 = + 1 1 - + 2 = 2 + 2 3 = 0 = 0

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