Finding the Locus of z for R: A Real Number

[Mentallic]

Homework Statement

If R is a real number, find the locus of z defined by:

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

Homework Equations

$$z=x+iy$$

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...

 

[gabbagabbahey]

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

 

[Mentallic]

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