If the real part of (z+1)/(z-1) is zero, find the locus of points representing z in the complex plane.
The answer in the book is x^2 + y^2 = 1 ie. a circle, but i dont know how to solve it
$\displaystyle \displaystyle \frac{z+1}{z-1} = \frac{x+1 + iy}{x-1 + iy}$
$\displaystyle \displaystyle = \frac{(x+1+iy)(x-1-iy)}{(x-1+iy)(x-1-iy)}$
$\displaystyle \displaystyle = \frac{(x+1+iy)(x-1-iy)}{(x-1)^2 + y^2}$
$\displaystyle \displaystyle = \frac{x^2 + y^2 - 1 - 2iy}{(x-1)^2 + y^2}$
$\displaystyle \displaystyle = \frac{x^2 + y^2 - 1}{(x - 1)^2 + y^2} + i\left[\frac{-2y}{(x - 1)^2 + y^2}\right]$.
You know that the real part is 0, so $\displaystyle \displaystyle \frac{x^2 + y^2 - 1}{(x-1)^2 + y^2} = 0$
$\displaystyle \displaystyle x^2 + y^2 - 1 = 0$
$\displaystyle \displaystyle x^2 + y^2 = 1$.