Let α be a root of the equation a - c x 2 + b - a x + c - b = 0 where a , b , c are distinct real numbers such that the matrix α 2 α 1 1 1 1 a b c is singular. Then the value of a - c 2 b - a c - b + b - a 2 a - c c - b + c - b 2 a - c b - a is

Step-by-step solution

It is given that the matrix α 2 α 1 1 1 1 a b c is singular, so α 2 α 1 1 1 1 a b c = 0 ⇒ α 2 c - b - α c - a + b - a = 0 Above equation holds true if α = 1 . Now, a - c 2 b - a c - b + b - a 2 a - c c - b + c - b 2 a - c b - a = a - b 3 + b - c 3 + c - a 3 a - b b - c c - a = 3 a - b b - c c - a a - b b - c c - a = 3 ∵ x 3 + y 3 + z 3 = 3 x y z   if   x + y + z = 0