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{\bf Question}

In the Poinc\'{a}re disc \textbf{D}, consider the parallelogram
$P_s$ bounded by the four hyperbolic lines
$$\ell_1=\{z\in\textbf{D}:\textrm{Re}(z)=0\}$$
$$\ell_2=\{z\in\textbf{D}:|z-2i|=\sqrt{3}\}$$
$$\ell_3=\{z\in\textbf{D}:|z+2i|=\sqrt{3}\}$$ and
$$\ell_s=\{z\in\textbf{D}:|z-s|=\sqrt{s^2-1}\}$$ where$s$ is real
and $s>1$. Determine the values of $s$ for which $P_s$ has finite
hyperbolic area.
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{\bf Answer}

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$P_s$ has finite area when $\ell_s$ and $\ell_2$ (or $\ell_s$ and
$\ell_3$ by symmetry) are parallel but not ultraparallel (that is,
when $\ell_2\ell_s$ intersect at the boundary $S^1$ at infinity of
${\bf{D}}$).

Since this occurs when $\ell_2\ell_s$ are tangent, the distance
between their centers is equal to the sum of their radii, and so

$|s-2i|=\sqrt{3}+\sqrt{s^2-1}$

$\sqrt{s^2+4}=\sqrt{3}+\sqrt{s^2-1}$

\un{Squaring}:

$s^2+4=3+s^2-1+2\sqrt{3}\sqrt{s^2-1}$

$2=2\sqrt{3}\sqrt{s^2-1}$

$s^2-1=\ds\frac{1}{3},\ \ \ s^2=\ds\frac{4}{3},\ \ \
s=\ds\frac{2}{\sqrt{3}}$ and so $P_s$ has finite hyperbolic area
for $s \geq \ds\frac{2}{\sqrt{3}}$.

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