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JEE Main202325 Jan 2023Morning ShiftMathematicsThree Dimensional GeometryActual

Let the equation of the plane passing through the line x - 2 y - z - 5 = 0 = x + y + 3 z - 5 and parallel to the line x + y + 2 z - 7 = 0 = 2 x + 3 y + z - 2 be a x + b y + c z = 65 . Then the distance of the point a , b , c from the plane 2 x + 2 y - z + 16 = 0 is _____ .

Correct answer

0

Step-by-step solution

Equation of plane through the line x - 2 y - z - 5 = 0 = x + y + 3 z - 5 is x - 2 y - z - 5 + λ x + y + 3 z - 5 = 0 ⇒ 1 + λ x + - 2 + λ + - 1 + 3 λ z + - 5 - 5 λ = 0 Also, this plane is parallel to the line x + y + 2 z - 7 = 0 = 2 x + 3 y + z - 2 , so 1 + λ - 2 + λ - 1 + 3 λ 1 1 2 2 3 1 = 0 ⇒ - 5 1 + λ + 3 λ - 2 + 3 λ - 1 = 0 ⇒ - 5 - 5 λ + 3 λ - 6 + 3 λ - 1 = 0 ⇒ λ = 12 So, equation of plane is 13 x + 10 y + 35 z = 65

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