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JEE MainMathematicsThree Dimensional Geometry

Let the lines L₁: x-3 2 = y-5 3 = z-7 and L₂: x-2 1 = y-4 2 = z-5 3 intersect at the point P( , , ) . If L₃ and L₄ are the lines given by x- 1 = y- -2 = z- 2 and x- 2 = y-2 1 = z-1 2 respectively, then the shortest distance between L₃ and L₄ is equal to:

Options

  1. A29 65
  2. B31 65
  3. C149 3
  4. D19 65

Correct answer

A. 29 65

Step-by-step solution

Let the lines be L₁ and L₂ . Any point on L₂ can be written as (r+2, 2r+4, 3r+5) . Since L₁ and L₂ intersect, this point must satisfy the equations of L₁ for some value of r : r+2-3 2 = 2r+4-5 3 r-1 2 = 2r-1 3 3r - 3 = 4r - 2 r = -1 Substituting r = -1 , the point of intersection is P(1, 2, 2) . Thus, = 1, = 2, = 2 . To find , we use the third part of the L₁ equation: 2-7 = -1-1 2 = -1 -5 = -1 = 5 Now, we substitute these values into the equations of L₃ and L₄ : L₃: x-1 1 = y-2 -2 = z-2 2 L₄: x-5 2 = y-2 1 = z-1 2

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