Argand Diagram: Difference in argument = pi/4

[unscientific]

Homework Statement

Sketch the locus of points in the argand diagram defined by z:

arg (z-1) - arg(z+1) = ∏/4

Homework Equations

The Attempt at a Solution

By simple geometry i worked out that at a point in the x-y plane, the angle subtended from that point to -1 and 1 must be = pi/4.

For a circle I know this must be = pi/2. But for pi/4 I have no clue..

 

[LCKurtz]

Homework Statement

Sketch the locus of points in the argand diagram defined by z:

arg (z-1) - arg(z+1) = ∏/4

Homework Equations

The Attempt at a Solution

By simple geometry i worked out that at a point in the x-y plane, the angle subtended from that point to -1 and 1 must be = pi/4.

For a circle I know this must be = pi/2. But for pi/4 I have no clue..

If $z=x+yi$ and you call $\theta_1=arg(z-1)$and $\theta_2=arg(z+1)$ what happens if you calculate $\tan(\theta_1-\theta_2)$?

 

[Dick]

By simple geometry i worked out that at a point in the x-y plane, the angle subtended from that point to -1 and 1 must be = pi/4.

For a circle I know this must be = pi/2. But for pi/4 I have no clue..

I know that it's a different circle, but that's for me you know and you to find out after you put some work in on this. How do you know pi/2 defines a circle? Apply the same ideas.