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JEE Main202312 Apr 2023Morning ShiftMathematicsThree Dimensional GeometryActual

Let the plane P : 4 x - y + z = 10 be rotated by an angle π 2 about its line of intersection with the plane x + y - z = 4 . If α is the distance of the point ( 2 , 3 , - 4 ) from the new position of the plane P , then 35 α is equal to

Options

  1. A85
  2. B105
  3. C126
  4. D90

Correct answer

C. 126

Step-by-step solution

We know that if a plane passes through the line of intersection of planes P 1 and P 2 then the family of planes is given by P 1 + λ P 2 where λ ∈ R . Equation of plane after rotation : 4 x - y + z - 10 + λ x + y - z - 4 = 0 ⇒ 4 + λ x + y λ - 1 + z 1 - λ - 4 λ - 10 = 0 Now angle between the new plane and 4 x - y + z - 10 is 90 ° . ⇒ n → 1 · n → 2 = 0 ⇒ 4 + λ 4 + λ - 1 - 1 + 1 - λ 1 = 0 ⇒ 16 + 4 λ - λ + 1 +

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