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JEE MainMathematicsThree Dimensional Geometry

Let L₁: x-4 1 = y-2 -1 = z 0 and L₂: x 3 = y 1 = z 1 be two lines. Let L₃ be a line passing through the point P( , , ) and perpendicular to both L₁ and L₂ . If L₃ intersects L₁ , then the minimum possible value of ^2 + ^2 + ^2 is equal to :

Options

  1. A16
  2. B4
  3. C2
  4. D12

Correct answer

A. 16

Step-by-step solution

The direction ratios of L₁ are 1, -1, 0 and of L₂ are 3, 1, 1 . Since L₃ is perpendicular to both L₁ and L₂ , its direction vector d₃ is: d₃ = vmatrix i & j & k 1 & -1 & 0 3 & 1 & 1 vmatrix = -1 i - 1 j + 4 k Let the point of intersection of L₃ and L₁ be A(k+4, -k+2, 0) . Since P( , , ) lies on L₃ , its coordinates can be written parametrically as: = k + 4 - = -k + 2 - = 4 To find the locus of P , we eliminate k and . Adding the first two equations gives: + = 6 - 2 From the third equation, = 4 . Substituting this i

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