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JEE MainMathematicsThree Dimensional Geometry

Let the line L₁ passing through the point (k, 1, 2) and parallel to the vector i + 2 j + 3 k intersect the line L₂ : x-1 2 = y-4 1 = z-2 -3 at the point P . If O is the origin, then the value of k^2 + OP^2 is

Options

  1. A33
  2. B13
  3. C35
  4. D39

Correct answer

D. 39

Step-by-step solution

The equation of line L₁ is x-k 1 = y-1 2 = z-2 3 = . Any point on L₁ is ( + k, 2 + 1, 3 + 2) . The equation of line L₂ is x-1 2 = y-4 1 = z-2 -3 = . Any point on L₂ is (2 + 1, + 4, -3 + 2) . For the lines to intersect at point P , their coordinates must be equal for some and : + k = 2 + 1 ... (1) 2 + 1 = + 4 2 - = 3 ... (2) 3 + 2 = -3 + 2 + = 0 ... (3) Solving (2) and (3), we get = 1 and = -1 . Substituting = -1 into the coordinates of L₂ , the intersection point is P(-1, 3, 5) . Substituting = 1 and = -1 into (1):

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