under what conditions $|z-a|+|z-b|=c$ and $|b-a| \gt c$ Represents Hyperbola

Question

under what conditions $|z-a|+|z-b|=c$ and $|b-a| \gt c$ Represents Hyperbola

Suppose if i take $$|z-1|+|z+1|=0.5$$ then we have

$$|z-1|=0.5-|z+1|$$

Squaring both sides

$$|z|^2+1-2Re(z)=0.25+|z|^2+1+2Re(z)-|z+1|$$ So

$$4x+0.25=\sqrt{(x+1)^2+y^2}$$ again squaring we get

$$16x^2-\frac{16}{15}y^2=1$$ which is Hyperbola

so is it true if $a,b \in \mathbb{C}$ and $|b-a| \gt c$ then $$|z-a|+|z-b|=c$$ represents hyperbola