For the following two (02) items: Consider the equation abx^2 + bcx + ca = cax^2 + abx + bc . If the roots of the equation are equal, then a , b , c are in

Step-by-step solution

The given equation is abx^2 + bcx + ca = cax^2 + abx + bc . Rearranging the terms, we get: (ab - ca)x^2 + (bc - ab)x + (ca - bc) = 0 a(b - c)x^2 + b(c - a)x + c(a - b) = 0 The sum of the coefficients is: a(b - c) + b(c - a) + c(a - b) = ab - ac + bc - ab + ac - bc = 0 Since the sum of the coefficients is 0 , one root of the equation is x = 1 . It is given that the roots of the equation are equal, so the other root must also be x = 1 . The product of the roots is given by c(a - b) a(b - c) . Equating the product of