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JEE Main20241 Feb 2024Evening ShiftMathematicsThree Dimensional GeometryActual

If the mirror image of the point P ( 3 , 4 , 9 ) in the line x − 1 3 = y + 1 2 = z − 2 1 is α , β , γ , then 14 α + β + γ is:

Options

  1. A102
  2. B138
  3. C108
  4. D132

Correct answer

C. 108

Step-by-step solution

The given point is P ( 3 , 4 , 9 ) . Let, Q α , β , γ be the image of P ( 3 , 4 , 9 ) and ' O ' be the mid-point of P Q which is on the given line x - 1 3 = y + 1 2 = z - 2 1 = λ (say) ⇒ x = 3 λ + 1 , y = 2 λ - 1 , z = λ + 2 let 3 λ + 1 , 2 λ - 1 , λ + 2 be the co-ordinates of O . Direction ratios of the line is a 1 = 3 , b 1 = 2 , c 1 = 1 Also, direction ratios of P O are 3 λ - 2 , 2 λ - 5 , λ - 7 As, P O is perpendicular to the line 3 3 λ - 2 + 2 2 λ - 5 + λ - 7 = 0 ⇒ 14 λ - 23 = 0 ⇒ λ = 23 14 ∴ O point is 3 · 23

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