Let α , β , γ be three real numbers satisfying α β γ 2 - 1 1 - 1 - 1 - 2 - 1 2 1 = 0 0 0 . If the point A α , β , γ lies on the plane 2 x + y + 3 z = 2 , then 3 α + 2 β - 6 γ is equal to

Step-by-step solution

α β γ 2 - 1 1 - 1 - 1 - 2 - 1 2 1 = 0 0 0 2 α - β - γ - α - β + 2 γ α - 2 β + γ = 0 0 0 ⇒ 2 α - β - γ = 0 - α - β + 2 γ = 0 α - 2 β + γ = 0 Using cramer’s rule ∆ = 2 - 1 - 1 - 1 - 1 2 1 - 2 1 = 2 - 1 + 4 + 1 - 1 - 2 - 1 2 + 1 = 6 - 3 - 3 = 0 So, the system of equations has infinite solutions Put γ = k 2 α - β = k α + β = 2 k ⇒ γ = k , β = k ⇒ α , β , γ ≡ k , k , k ∵ α , β , γ satisfies 2 x + y + 3 z = 2 ⇒ 6 k = 2 ⇒ k = 1 3 α , β , γ ≡ 1 3 , 1 3 , 1 3 ⇒ 3 α + 2 β - 6 γ = - 1 3