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JEE Main2016MathematicsThree Dimensional GeometryActual

The shortest distance between the lines x 2 = y 2 = z 1 and x + 2 - 1 = y - 4 8 = z - 5 4 , lies in the interval:

Options

  1. A3 , 4
  2. B2 , 3
  3. C1 , 2
  4. D0 , 1

Correct answer

B. 2 , 3

Step-by-step solution

Let the direction vectors of the two lines be given by- a 1 i ^ + b 1 j ^ + c 1 k ^ = 2 i ^ + 2 j ^ + k ^ and a 2 i ^ + b 2 j ^ + c 2 k ^ = - i ^ + 8 j ^ + 4 k ^ The shortest distance between the two lines is given by- x 2 - x 1 i ^ + y 2 - y 1 j ^ + z 2 - z 1 k ^ . a 1 i ^ + b 1 j ^ + c 1 k ^ × a 2 i ^ + b 2 j ^ + c 2 k ^ a 1 i ^ + b 1 j ^ + c 1 k ^ × a 2 i ^ + b 2 j ^ + c 2 k ^ = x 2 − x 1 y 2 -  y 1 z 2 − z 1 a 1 b 1 c 1 a 2 b 2 c 2 i j k a 1 b 1 c 1 a 2 b 2 c 2 = − 2 4 5 2 2 1

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