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JEE Main202313 Apr 2023Morning ShiftMathematicsThree Dimensional GeometryActual

Let the equation of plane passing through the line of intersection of the planes x + 2 y + a z = 2 and x − y + z = 3 be 5 x − 11 y + b z = 6 a − 1 . For c ∈ ℤ , if the distance of this plane from the point a , − c , c is 2 a , then a + b c is equal to

Options

  1. A2
  2. B4
  3. C- 4
  4. D- 2

Correct answer

C. - 4

Step-by-step solution

Let the equation of plane passing through the intersection of two given planes: x + 2 y + a z - 2 + λ ( x - y + z - 3 ) = 0 ⇒ x ( λ + 1 ) + y ( 2 - λ ) + z ( a + λ ) - 2 - 3 λ = 0 This is same as 5 x - 11 y + b z = 6 a - 1 λ + 1 5 = 2 - λ - 11 = a + λ b = 2 + 3 λ 6 a - 1 ⇒ - 11 λ - 11 = 10 - 5 λ ⇒ 6 λ = - 21 ⇒ λ = - 7 2 2 - λ - 11 = 2 + 3 λ 6 a - 1 ⇒ 2 + 7 2 - 11 = 2 - 21 2 6 a - 1 ⇒ - 1 2 = - 17 2 ( 6 a -

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