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JEE Main202310 Apr 2023Morning ShiftMathematicsThree Dimensional GeometryActual

Let two vertices of a triangle A B C be 2 , 4 , 6 and 0 , - 2 , - 5 , and its centroid be 2 , 1 , - 1 . If the image of the third vertex in the plane x + 2 y + 4 z = 11 is α , β , γ , then α β + β γ + γ α is equal to

Options

  1. A70
  2. B76
  3. C74
  4. D72

Correct answer

C. 74

Step-by-step solution

Given, Two vertices of a triangle A B C be 2 ,   4 ,   6 and 0 , - 2 , - 5 , and its centroid be 2 , 1 , - 1 Now, let the vertex ' C ' be a ,   b ,   c and using centroid formula we get, 2 = 2 + 0 + a 3 ⇒ a = 4 1 = 4 - 2 + b 3 ⇒ b = 1 - 1 = 6 - 5 + c 3 ⇒ c = - 4 So, the third vertex will be, C 4 ,   1 ,   - 4 Now finding image of C in x + 2 y + 4 z = 11 we get, α - 4 1 = β - 1 2 = γ + 4 4 = - 2 4 + 2 - 16 - 11 16 + 4 + 1 ⇒ α - 4 1 = 2

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