Describing a set of elements in a complex plane

Question

Let $z_1$,$z_2$ e $\mathbb C$, $z_1 \not= z_2$ be two points in the complex plane.

Describe the set $S_1$ $=$ {$z$ e $\mathbb C$: $(z-z_1)^2$ + $(z-z_2)^2$ = $(z_1-z_2)^2$}

My attempt:

I expanded the above using remarkable identities, and got that:

$z^2-zz_1-zz_2 = - z_1z_2$ $z^2 - z(z_1+z_2)+z_1z_2=0$ Thus $z_1$ and $z_2$ are two distinct roots of the above quadratic equation, thus $z=z_1$ or $z=z_2$

Is it correct?

What happens if the parentheses were replaced by absolute value? I mean the set becomes: $S_2$ = {$z$ e $\mathbb C$: |$z-z_1|^2$ + $|z-z_2|^2$ = $|z_1-z_2|^2$}

Answer 1

Your answer t the first is correct and in fact clever. By Pythagorous Theorem $S_2$ consists of all $z$ such that angle at $z$ in the triangle formed by $z,z_1,z_2$ is $\pi /2$.