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COMEDK2025MathematicsThree Dimensional GeometryActual

Two lines x-1 2 = y+1 3 = z-1 4 and x-3 1 = y-k 2 = z 1 intersect at a point. Then the value of ' k ' is

Options

  1. A- 13 2
  2. B13 2
  3. C7 2
  4. D9 2

Correct answer

D. 9 2

Step-by-step solution

Let the first line be L₁: x-1 2 = y+1 3 = z-1 4 = . Any point on L₁ is P(2 + 1, 3 - 1, 4 + 1) . Let the second line be L₂: x-3 1 = y-k 2 = z 1 = . Any point on L₂ is Q( + 3, 2 + k, ) . Since the lines intersect, there exist and such that P = Q . Equating the coordinates: 1) 2 + 1 = + 3 2 - = 2 2) 3 - 1 = 2 + k 3 - 2 = k + 1 3) 4 + 1 = 4 - = -1 Subtracting equation (1) from equation (3): (4 - ) - (2 - ) = -1 - 2 2 = -3 = - 3 2 Substituting = - 3 2 into equation (1): 2(- 3 2 ) - = 2 -3 - = 2 = -5 Substituting = - 3 2

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