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{\bf Question}

Let A be the Euclidean circle in \textbf{H} with Euclidean centre
$1+2i$ and Euclidean radius 1. We know from class that A is also a
hyperbolic circle. Determine its hyperbolic centre and hyperbolic
radius.

\medskip

{\bf Answer}

The hyperbolic center is the midpoint of the hyperbolic line
segment joining two points on a hyperbolic diameter, such as $1+i$
and $1+3i$.  Since $d_{\bf{H}}(1+i,1+3i)=\ln(3)$, the hyperbolic
center is the point $1+ai$ which satisfies

$$d_{\bf{H}}(1+i,1+ai)=\ds\frac{1}{2}\ln(3)=d_{\bf{H}}(1+ai,1+3i)$$

$$\ln(a)=ds\frac{1}{2}\ln(3)=\ln\left(\ds\frac{3}{a}\right)=\ln(3)-\ln(a)$$

\un{So, $a=\sqrt{3}$} and so the hyperbolic center is
\un{$1+\sqrt{3}i$}.

The hyperbolic radius is
$d_{\bf{H}}(1+i,1+\sqrt{3}i)=\ds\frac{1}{2}\ln(3)$.

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