# Loci for complez numbers

1. Jul 11, 2008

### rock.freak667

1. The problem statement, all variables and given/known data
If arg(z-1)=arg(z+1)=$\frac{3 \pi}{4}$, find the locus of z.

2. Relevant equations

3. The attempt at a solution

arg(z-1)=arg(z-[1+0i]) => z lies on half the line through the point (1,0) excluding (1,0), inclined at alpha

Similarly arg(z+1) =>z lies on half the line through (-1,0),excluding (-1,0).Inclined at beta.

If I draw those two on the same graph, they form a triangle. But how do I incorporate the 3pi/4 ?

or was I supposed to draw arg(z-1)=3pi/4?

2. Jul 11, 2008

### tiny-tim

Hi rock.freak667!
That doesn't look right to me.

Are you sure it isn't arg(z-1) - arg(z+1) = 3π/4 ?

3. Jul 11, 2008

### rock.freak667

Well it actually could be that because it's in someone's handwriting.

But if it was arg(z-1) - arg(z+1) = 3π/4.

How would I go about it?

EDIT:

I can sketch

$$arg(z-z_0)= \lambda$$

where z_0 is a fixed complex number and lambda is the argument.

Last edited: Jul 11, 2008
4. Jul 11, 2008

### tiny-tim

dot-product or just trigonometry?

5. Jul 11, 2008

### rock.freak667

ahh nevermind I figured it out.

And the correct question is:

$$arg(z-1)-arg(z+1)=\frac{\pi}{4}$$