If a , b , c are non-zero real numbers, the system of equations y + z = a + 2 x , x + z = b + 2 y , x + y = c + 2 z is consistent and b = 4 a + c 4 , then the sum of the roots of the equation a t 2 + b t + c = 0 is

Step-by-step solution

Given, 2 x - y - z = - a x - 2 y + z = b x + y - 2 z = c For consistent system of equations ∆ ≠ 0 or ∆ = ∆ 1 = ∆ 2 = ∆ 3 = 0 ∆ = 2 - 1 - 1 1 - 2 1 1 1 - 2 = 0 ∆ 1 = - a - 1 - 1 b - 2 1 c 1 - 2 = 0 ⇒ a + b + c = 0 ∆ 2 = 2 - a - 1 1 b 1 1 c - 2 = 0 ⇒ a + b + c = 0 ∆ 3 = 2 - 1 - a 1 - 2 b 1 1 c = 0 ⇒ a + b + c = 0 a + b + c = 0 16 a - 4 b + c = 0 ⇒ a 5 = b 15 = c - 20 ⇒ a = λ b = 3 λ c = - 4 λ ⇒ Sum of roots = - b a = - 3