Sketching points given by complex numbers

Question

I cannot remember much about circles. If I have $|z-1+i|=1$, how do I translate this geometrically. I know it's a circle but I can't remember how to do this.

Answer 1

We have

$$|z+i-1|=1.$$

if $z=x+iy$ we then have

$$|x+iy+i-1|=1,$$

$$|(x-1)+i(y+1)|=1.$$

Recalling the definition of $|z|$ we have

$$\sqrt{(x-1)^2+(y+1)^2}=1,$$

$$\implies (x-1)^2+(y+1)^2=1.$$

Thus, $|z+i-1|=1$ is a circle centered at $(1,-1)$ of radius one.