If a line passing through - 2 , 1 , α and 4,1 , 2 is perpendicular to the vector 3 i ^ - 4 j ^ + 5 k ^ and parallel to the plane containing vectors i ^ + 2 β k ^ and 2 β j ^ + α k ^   ∀ β ≠ 0 , then 10 α + β is equal to

Step-by-step solution

Let, coordinates of A , B are respectively - 2,1 , α , 4,1 , 2 respectively A B → = 6 i ^ + 2 - α k ^ ∵ A B → is perpendicular to 3 i ^ - 4 j ^ + 5 k ^ ⇒ 18 + 5 2 - α = 0 ⇒ α = 28 5 Also, A B → , i ^ + 2 β k ^ , 2 β j ^ + α k ^ are coplanar ⇒ 6 0 2 - α 1 0 2 β 0 2 β α = 0 ⇒ 2 β 12 β + α - 2 = 0 ⇒ β = 0 or 12 β = 2 - α 12 β = 2 - 28 5 12 β = - 18 5 ⇒ β = - 3 10