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JEE Main202431 Jan 2024Evening ShiftMathematicsThree Dimensional GeometryActual

The shortest distance between lines L 1 and L 2 , where L 1 : x − 1 2 = y + 1 − 3 = z + 4 2 and L 2 is the line passing through the points A − 4 , 4 , 3 , B − 1 , 6 , 3 and perpendicular to the line x − 3 − 2 = y 3 = z − 1 1 , is

Options

  1. A121 221
  2. B24 117
  3. C141 221
  4. D42 117

Correct answer

C. 141 221

Step-by-step solution

Given: L 2 passes through A - 4 , 4 , 3 and B - 1 , 6 , 3 . ⇒ L 2 : x + 4 3 = y − 4 2 = z − 3 0 Also, L 1 : x - 1 2 = y + 1 - 3 = z + 4 2 We know that, shortest distance between L 1 : x - x 1 a 1 = y - y 1 b 1 = z - z 1 c 1 and L 2 : x - x 2 a 2 = y - y 2 b 2 = z - z 2 c 2 is given by, x 2 - x 1 y 2 - y 1 z 2 - z 1 a 1 b 1 c 1 a 2 b 2 c 2 a 1 b 2 - a 2 b 1 2 + b 1 c 2 - b 2 c 1 2 + c 1 a 2 - c 2 a 1 2 . So, shortest distance, D = - 4 - 1 1 + 4 4 + 3 2 − 3 2 3 2 0 4 + 9 2 + 0 - 4 2 + 6 - 0 2 ⇒ D = - 5 5 7 2 − 3 2 3

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