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JEE Main202226 Jun 2022Morning ShiftMathematicsThree Dimensional GeometryActual

Let the plane 2 x + 3 y + z + 20 = 0 be rotated through a right angle about its line of intersection with the plane x - 3 y + 5 z = 8 . If the mirror image of the point 2 , - 1 2 , 2 in the rotated plane is B a , b , c , then

Options

  1. Aa 8 = b 5 = c - 4
  2. Ba 4 = b 5 = c - 2
  3. Ca 8 = b - 5 = c 4
  4. Da 4 = b 5 = c 2

Correct answer

A. a 8 = b 5 = c - 4

Step-by-step solution

Let the equation of the rotated plane be 2 x + 3 y + z + 20 + λ x - 3 y + 5 z - 8 = 0 2 + λ x + 3 - 3 λ y + 1 + 5 λ z + 20 - 8 λ = 0 This plane is perpendicular to 2 x + 3 y + z + 20 = 0 So, 2 2 + λ + 3 3 - 3 λ + 1 + 5 λ = 0 ⇒ λ = 7 So the equation of rotated plane will be x - 2 y + 4 z - 4 = 0 Also given that the mirror image of A 2 , - 1 2 , 2 in rotated plane is B a , b , c Equation of A B : x - 2 1 = y + 1 2 - 2 = z - 2 4 = k Let coordinate of B be 2 + k , - 1

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