Checking the answer to complex number question

[rock.freak667]

Homework Statement

Find an equation connecting x and y for which (z-1)/(z+1) has an argument $\alpha$

Homework Equations

z=x+iy

arg(z)=tan^-1(y/x)

The Attempt at a Solution

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

Substituting z=x+iy

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

Realizing

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

Re:i²=-1

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

Thus

$$tan\alpha = \frac{2y}{x^2+y^2-1}$$

Is this correct? Or should I just put in the form of a circle?

 

[tiny-tim]

Find an equation connecting x and y for which (z-1)/(z+1) has an argument $\alpha$
…
Thus

$$tan\alpha = \frac{2y}{x^2+y^2-1}$$

Is this correct? Or should I just put in the form of a circle

Hi rock.freak667!

Yes, that's messy!

Definitely put it in the form of a nice circle (x² + (y-a)² = b²)!