Let α , β , γ , δ be real numbers such that α 2 + β 2 + γ 2 ≠ 0 and α + γ = 1 . Suppose the point 3 , 2 , - 1 is the mirror image of the point 1 , 0 , - 1 with respect to the plane α x + β y + γ z = δ . Then which of the following statements is/are TRUE?

Step-by-step solution

From the diagram, Mid point of A & A ' lies on the plane, i.e., B 2 , 1 , - 1 2 α + β - γ = δ … 1 α + γ = 1 … . 2 Direction Ratio of AA ' 2 , 2 , 0 will be proportional to normal to the plane, α 2 = β 2 = γ 0 = λ α = 2 λ , β = 2 λ , γ = 0 ∴ 2 λ + 0 = 1 ⇒ λ = 1 2 ∴ α = 1 , β = 1 , γ = 0 , δ = 3 Now, (A) α + β + γ = 1 + 1 + 0 = 2 ≠ δ (B) α + β = 1 + 1 = 2 (C) δ + β = 3 + 1 = 4 (D) δ - γ = 3