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JEE Main202224 Jun 2022Evening ShiftMathematicsThree Dimensional GeometryActual

If the shortest distance between the lines x - 1 2 = y - 2 3 = z - 3 λ and x - 2 1 = y - 4 4 = z - 5 5 is 1 3 , then the sum of all possible values of λ is:

Options

  1. A16
  2. B6
  3. C12
  4. D15

Correct answer

A. 16

Step-by-step solution

We know that, shortest distance between two line is given by a 2 - a 1 · b 1 × b 2 b 1 × b 2 Here, a 1 = 1 , 2 , 3 a 2 = 2 , 4 , 5 b → 2 = 2 i ^ + 3 j ^ + λ k ^ and b → 2 = i ^ + 4 j ^ + 5 k ^ So, S.D. = 2 - 1 i ^ + 4 - 2 j ^ + 5 - 3 k ^ · b → 1 × b → 2 b 1 × b 2 Calculating b → 1 × b → 2 = i ^ j ^ k ^ 2 3 λ 1 4 5 = i ^ 15 - 4 λ + j ^ λ - 10 + k ^ 5 = 15 - 4 λ i ^ + λ - 10 j ^ + 5 k ^ So, b → 1 × b &#8594

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