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COMEDK2021MathematicsThree Dimensional Geometry

The point of intersection of the lines x-1 2 = y-2 3 = z-3 4 and x-4 5 = y-1 2 =z is

Options

  1. A(0,0,0)
  2. B(1,1,1)
  3. C(-1,-1,-1)
  4. D(1,2,3)

Correct answer

C. (-1,-1,-1)

Step-by-step solution

Let the first line be x-1 2 = y-2 3 = z-3 4 = s . Then the coordinates of any point on this line are (2s+1, 3s+2, 4s+3) . Let the second line be x-4 5 = y-1 2 = z 1 = t . Then the coordinates of any point on this line are (5t+4, 2t+1, t) . For the lines to intersect, there must exist values of s and t such that the coordinates are equal: 2s+1 = 5t+4 2s - 5t = 3 3s+2 = 2t+1 3s - 2t = -1 4s+3 = t Substituting t = 4s+3 into the first equation: 2s - 5(4s+3) = 3 2s - 20s - 15 = 3 -18s = 18 s = -1 . Substituting s = -1 i

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