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JEE Main20236 Apr 2023Morning ShiftMathematicsThree Dimensional GeometryActual

If the equation of the plane passing through the line of intersection of the planes 2 x - y + z = 3 , 4 x - 3 y + 5 z + 9 = 0 and parallel to the line x + 1 - 2 = y + 3 4 = z - 2 5 is a x + b y + c z + 6 = 0 , then a + b + c is equal to

Options

  1. A12
  2. B14
  3. C16
  4. D13

Correct answer

B. 14

Step-by-step solution

Given, The equation of the plane passing through the line of intersection of the planes 2 x - y + z = 3 ,   4 x - 3 y + 5 z + 9 = 0 and parallel to the line x + 1 - 2 = y + 3 4 = z - 2 5 is a x + b y + c z + 6 = 0 , Now let, P 1 = 2 x - y + z = 3 And P 2 = 4 x - 3 y + 5 z + 9 = 0 Now using family of planes we get, P 1 + λ P 2 = 0 ⇒ ( 2 x - y + z - 3 ) + λ ( 4 x - 3 y + 5 z + 9 ) = 0 ⇒ ( 2 + 4 λ ) x - ( 1 + 3 λ ) y + ( 1 + 5 λ ) z - ( 3 - 9 λ ) = 0 Now let, P 3 = ( 2 +

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