# Complex locus 2

1. Dec 12, 2008

### Mentallic

1. The problem statement, all variables and given/known data
If R is a real number, find the locus of z defined by:

$$z=\frac{1+iR}{1-iR}$$

2. Relevant equations
$$z=x+iy$$

3. The attempt at a solution
$$z=(\frac{1+iR}{1-iR})(\frac{1+iR}{1+iR})$$

$$=\frac{1-R^2+i2R}{1+R^2}$$

Therefore, $$x=\frac{1-R^2}{1+R^2}$$

and $$y=\frac{2R}{1+R^2}$$

I'm unsure what to do now though...

2. Dec 12, 2008

### gabbagabbahey

Hmmm... why not compute $$|z|$$ and see if that gives you anything useful

3. Dec 12, 2008

### Mentallic

Everything just fell into place and $$|z|=1$$. How did you know that would happen? lol