If the system of equations 2 x + y - z = 5 2 x - 5 y + λ z = μ x + 2 y - 5 z = 7 has infinitely many solutions, then ( λ + μ ) 2 + ( λ - μ ) 2 is equal to

Step-by-step solution

Given system of equations. 2 x + y - z = 5 2 x - 5 y + λ z = μ x + 2 y - 5 z = 7 For infinite solution Δ = 0 = Δ 1 = Δ 2 = Δ 3 ⇒ Δ = 2 1 - 1 2 - 5 λ 1 2 - 5 = 0 ⇒ 51 - 3 λ = 0 ⇒ λ = 17 ⇒ Δ 3 = 5 2 1 μ 2 - 5 7 1 2 = 0 ⇒ - 3 ( μ + 13 ) = 0 ⇒ μ = - 13 Now ( λ + μ ) 2 + ( λ - μ ) 2 = ( 17 + 13 ) 2 + ( 17 - 13 ) 2 = 900 + 16 = 916 Hence this is the correct option.