How can i prove $S=\{ z: \Re (z) \leq -a \text{ and} \Im{(z)}=0\}$?

Question

I'm trying to show that $S=\{ z: \Re (z) \leq -a \wedge \Im{(z)}=0\}$, when $S=\{z\in\mathbb{C}:|z-a|-|z+a|=2c\}$, $a\in \mathbb{R^+}$ and $a=c$.

i tried the following:

$2a=|z-a|-|z+a|=|a-z|-|z+a|\leq 2|z|$, also calculate that $\Im{(z)}=0$, when $c=a$. Then $2a\leq 2x^2\Rightarrow x\leq -a \, \vee x \geq a$.

Where am i wrong?.

Answer 1

Note additionally that $x\ge a\implies |x-a|-|x+a|\lt0$. So only $x\le -a$ works.