Complex numbers and loci

$\displaystyle \frac{z+1}{z-1} = \frac{x+1 + iy}{x-1 + iy}$

$\displaystyle = \frac{(x+1+iy)(x-1-iy)}{(x-1+iy)(x-1-iy)}$

$\displaystyle = \frac{(x+1+iy)(x-1-iy)}{(x-1)^2 + y^2}$

$\displaystyle = \frac{x^2 + y^2 - 1 - 2iy}{(x-1)^2 + y^2}$

$\displaystyle = \frac{x^2 + y^2 - 1}{(x - 1)^2 + y^2} + i\left[\frac{-2y}{(x - 1)^2 + y^2}\right]$ .

You know that the real part is 0, so $\displaystyle \frac{x^2 + y^2 - 1}{(x-1)^2 + y^2} = 0$

$\displaystyle x^2 + y^2 - 1 = 0$

$\displaystyle x^2 + y^2 = 1$ .