Complex Analysis: Locus Sketching

[altcmdesc]

Homework Statement

Sketch the locus of |z-2i|=z+3 in C

2. The attempt at a solution

Let z=x+iy, then |z-i|=|x+iy-2i)|=|x+i(y-2)|=(x^2+(y-2)^2)^(1/2)=z+3

The problem is that I can't tell what this means geometrically. Is it a spiral?

 

[tiny-tim]

Sketch the locus of |z-2i|=z+3 in C

Hi altcmdesc!

Hint: |z - 2i| is real, so z + 3 must be real, so … ?

 

[altcmdesc]

So it must be the real line?

 

[tiny-tim]

So it must be the real line?

It must be part of the real line.

 

[altcmdesc]

I'm thinking of this as the simpler problem z=|z-i| first, but I'm having a hard time believing this could be true for any z because of the triangle inequality.

 

[tiny-tim]

I'm thinking of this as the simpler problem z=|z-i| first, but I'm having a hard time believing this could be true for any z because of the triangle inequality.

That's right!

But how about z+1=|z-i| ?