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NTA Abhyas JEE Main2020MathematicsThree Dimensional GeometryPractice

The plane 4 x + 7 y + 4 z + 81 = 0 is rotated through a right angle about its line of intersection with the plane 5 x + 3 y + 10 z = 25 . If the equation of the plane in its new position is x - 4 y + 6 z = K , then the value of K is

Options

  1. A106
  2. B- 89
  3. C73
  4. D37

Correct answer

A. 106

Step-by-step solution

The equation of the plane through the line of intersection of the planes is 4 x + 7 y + 4 z + 81 + λ 5 x + 3 y + 10 z - 25 = 0 4 + 5 λ x + 7 + 3 λ y + 4 + 10 λ z + 81 - 25 λ = 0 Which is parallel to x - 4 y + 6 z = K ⇒ 4 + 5 λ 1 = 7 + 3 λ - 4 = 4 + 10 λ 6 = 81 - 25 λ - K So, - 16 - 20 λ = 7 + 3 λ 23 λ = - 23 λ = - 1 and 81 - 25 λ - K = 4 + 10 λ 6 - K = 81 - 25 λ 6 4 + 10 λ = - 106 Hence, K = 106

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