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Let L₁: x-1 1 = y-2 -1 = z-3 -1 and L₂: x-1 1 = y-1 0 = z-1 be two lines. Let L₃ be a line passing through the point ( , , ) and perpendicular to both L₁ and L₂ . If L₃ intersects L₁ and all such points ( , , ) satisfy the relation 2 - + 3 = C , then the value of + C is equal to :

Options

  1. A5
  2. B4
  3. C8
  4. D13

Correct answer

D. 13

Step-by-step solution

The direction ratios of L₁ are 1, -1, -1 and of L₂ are 1, 0, . Since L₃ is perpendicular to both L₁ and L₂ , its direction vector d₃ is given by the cross product of their direction vectors: d₃ = d₁ d₂ = vmatrix i & j & k 1 & -1 & -1 1 & 0 & vmatrix = - i - ( + 1) j + k Any point ( , , ) on L₃ that intersects L₁ lies in the plane containing L₁ and parallel to L₃ . The normal n to this plane is perpendicular to both L₁ and L₃ : n = d₁ d₃ = vmatrix i & j & k 1 & -1 & -1 - & -( + 1) & 1 vmatrix n = (-1 - ( + 1)) i - (

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