Given two points and radius of circle, find centre and radius of another circle

Question

(Here, $\mathbb{H}^2$ is the hyperbolic plane.)

Let $\vec{u}=\left(-\frac35, \frac45\right)$, $\vec{b} = \left(\frac65, \frac25\right)$, $s=\sqrt{\|\vec{b}\|^2-1}$. Let $C$ be the Eucledian circle in $\mathbb{R}^2$ centred at $\vec{b}$ of radius $s$, and $L=C\cap \mathbb{H}^2$. Find the centre $\vec{a}\in\mathbb{R}^2$ and the radius $r$ of the Eucledian circle $D$ in $\mathbb{R}^2$ such that $M=D\cap \mathbb{H}^2$ is the hyperbolic line which is asymptotic to $\vec{u}$ and intersects orthogonally with $L$.

Here's my sketch of the problem:

I have come up with two equations:

$$\textbf{(1) } \|\vec{a}-\vec{u}\|^2 = r^2$$ $$\textbf{(2) } \text{Orthogonality condition for two circles}$$

But I have a hard time figuring out what a third equation could be. Would appreciate some advice.

Answer 1

I suppose you are working with the Poincare disk model ( please check in Wikipedia)

And you have to find the details (let's call them that) of the hyperbolic line which is asymptotic to  $u$ and intersects orthogonal with $L$

The hyperbolic line that you are looking for is not only orthogonal to $L$ it must also be orthogonal to the unit circle (otherwise it is just no hyperbolic line)

The euclidean centre of the euclidean circle you are looking for is on the intersection of:

• the euclidean line tangent to the unit circle going through $u$
• the euclidean line going to (both) intersections of the circle $C$ and the unit circle .

With the above you can calculate point $ a $ and also its radius .

Good luck

Ps there is a mistake in your drawing the centre $a$ is outside the unit disk