Complex Mappings: Branch of hyperbola mapped onto a line?

Question

I understand the mechanical method of converting mappings to parametric form, bu why does it say that a branch of the hyperbola gets mapped onto a line? From what I know about hyperbolas, $x^2 -y^2=c_1$ such that $c_1>0$ would trace out a hyperbola in a graph (i.e. as $x$ and $y$ varies, $c_1$ varies accordingly). Since $c_1$ varies(i.e. $c_1$ is not a constant real number), why would $x^2-y^2$ get mapped onto a line $c_1$ as mentioned in the textbook I'm reading?

Answer 1

What the text says is that for each fixed number $c_1>0$, the map $w=z^2$ maps the hyperbola $x^2-y^2=c_1$ onto the line $x=c_1$. That's so because, if $(x,y)$ belongs to that hyperbola, $\operatorname{Re}\bigl((x+y)^2\bigr)=x^2-y^2=c_1$. Besides, it's not hard to see that $\operatorname{Im}\bigl((x+y)^2\bigr)=2xy$ can take any real valu, so we do get the whole line $x=c_1$.