Graph of $|z+w| = 3$ when $z$ is any complex number and $w = 2+i$

Question

The battle with math continues. I would like to understand why the graph of $|z+w|=3$ is a circle of radius $3$ centered at $-w$. $z$ is any complex number and $w=2+i$. I get why it's a circle, but not how to find the center of it.

I don't know how to handle when one complex is just any complex.

Please try keep the explanation as simple as possible, I'm math stupid. I know about polar, exponential and rectangular forms if that helps in any way.

Thank you

Answer 1

Note that $|z+w|$ is the same as $|z-(-w)|$, which you ought to recognize as the distance between $z$ and $-w$.

So you're looking at the set of points $z$ whose distance to $-w$ is $3$, or in other words the circle with center $-w$ and radius $3$.

Answer 2

Write $z=x+iy$; thus

$$|x+iy+2+i|^2=9 \Longleftrightarrow (x+2)^2+(y+1)^2=9.$$

So the graph of $|z+w|=3$ is the circumference of center $(-2,-1)$ (after the canonical identification of $\Bbb C$ with $\Bbb R^2$, so the center is the point $-w\in \Bbb C$) and radius $3$.