Complex numbers and loci

**kumquat** · February 9th 2011, 05:22 AM

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

**Prove It** · February 9th 2011, 05:38 AM

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

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

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

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

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

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

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

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

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