The value of $m$ for which the equation $\cfrac{a}{x+a+m}+\cfrac{b}{x+b+m}=1$, has roots equal in magnitude but opposite in sign is

For roots to be equal in magnitude but opposite in sign, the sum of roots must be 0. The equation is a/(x+a+m) + b/(x+b+m) = 1. Rearranging: a(x+b+m) + b(x+a+m) = (x+a+m)(x+b+m). x(a+b) + ab + am + bx + ab + bm = x^2 + x(b+m) + x(a+m) + (a+m)(b+m). x(a+b) + 2ab + m(a+b) = x^2 + x(a+b+2m) + (a+m)(b+m). x^2 + x(2m) + (a+m)(b+m) - 2ab - m(a+b) = 0. For sum of roots to be 0, the coefficient of x must be 0, so 2m = 0, thus m = 0.