If 1,2 , p ,   2,8 , - 6 and α 2 - 2 α , p , 1 are ordered triplet pair of the form x , y , z which satisfy all the equations x a + y b + z c = 1 ,   x b + y c + z a = 1 and x c + y a + z b = 1 , then the sum of all the values of α is equal to (where, a b + b c + c a ≠ 0 )

Step-by-step solution

System of equation has more than one solution ∆ = 1 a 1 b 1 c 1 b 1 c 1 a 1 c 1 a 1 b = 0 ⇒ 1 a = 1 b = 1 c ⇒ a = b = c ⇒ all three equations are x + y + z = a Now, 2,8 , - 6 satisfy this equation 2 + 8 - 6 = a ⇒ a = 4 1,2 , p also satisfy this equation ⇒ 1 + 2 + p = 4 ⇒ p = 1 α 2 - 2 α , p , 1 also satisfy this equation ⇒ α 2 - 2 α + 1 + 1 = 4 ⇒ α 2 - 2 α - 2 = 0 Sum of the values of α = 2