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JEE MainMathematicsThree Dimensional Geometry

Consider the lines L₁ and L₂ given by L₁ : x - 2 1 = y -1 = z - 3 2 L₂ : x - 4 1 = y - 3 -2 = z 3 A line passing through the point P(2, 3, -1) intersects L₁ and L₂ at the points A and B respectively. Then the value of (AB)^2 is

Options

  1. A36
  2. B130
  3. C49
  4. D81

Correct answer

A. 36

Step-by-step solution

Let the points on L₁ and L₂ be A and B respectively. A = ( + 2, - , 2 + 3) B = ( + 4, -2 + 3, 3 ) Since the line passes through P(2, 3, -1) , the points A , P , and B are collinear. Thus, the direction ratios of PA and PB must be proportional. Direction ratios of PA = ( , - - 3, 2 + 4) Direction ratios of PB = ( + 2, -2 , 3 + 1) + 2 = - - 3 -2 = 2 + 4 3 + 1 Taking the first two parts and cross-multiplying: -2 = ( + 2)(- - 3) - 3 - 2 - 6 = 0 ... (i) Taking the first and third parts and cross-multiplying: (3 + 1) = (

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