Euclidean circle in complex plane

Question

I am reading Anderson's Hyperbolic Geometry and am having trouble with one of the Exercises in Chapter 1:

Consider the unit circle $\mathbb{S}^1=\{z \in \mathbb{C} \text{ s.t. }|z|=1\}$. Let $A$ be a Euclidean circle in $\mathbb{C}$ with Euclidean center $re^{i\theta}$, $r>1$, and Euclidean radius $s>0$. Show that $A$ is perpendicular to $\mathbb{S}^1$ if and only if $s=\sqrt{r^2-1}$.

I'm having trouble understanding the setup. For instance, what does it mean for the Euclidean circle to have center $re^{i\theta}$? I'm having trouble visualizing that. If someone can give some pointers about how to visualize this problem, I think I can figure it out.

Answer 1

Its all back to Pythagoras:

(the question is just to you all way trough it)

so a bit analytical:

there is:

• an origin point $O$
• a point at distance r (Lets call it $R$ )
• a circle $\mathbb{S}^1=$ with centre $O$ and radius $1$ (the unit circle)
• a circle $A$ with centre $R$ and radius $s= \sqrt{r^2-1} $

supose one of the intersections of the two circles is $C$

What size is the angle $\angle OCR$ ?

(make the triangle $\triangle OCR$ )