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\begin{document}

{\bf Question}

Let \textbf{A} be the Euclidean circle given by the equation
$$3z\overline{z}+10iz-10i\overline{z}+4=0$$ and let
$$m(z)=\frac{1}{z-1}.$$ Determine whether $m(\textbf{A})$ is a
Euclidean circle or the union of a Euclidean line with
$\{\infty\}$. In the former case, determine its Euclidean centre
and Euclidean radius. In the latter case, give its slope and the
$y$-intercept.
\medskip

{\bf Answer}

Determine the equation for $m(A)$: [15 points]

set $w=m(z)=\ds\frac{1}{z-1}$ and solve for {z}, so that
$z=1+\ds\frac{1}{w}$.

Substitute into the equation for $a$ and simplify:

\begin{eqnarray*} 0 & = & 3z\bar{z}+10iz-10i\bar{z}+4\\ & = &
3\left(\ds\frac{1}{w}+1\right)\left(1+\ds\frac{1}{\bar{w}}\right)
+10i\left(\ds\frac{1}{w}+1\right)-10i\left(\ds\frac{1}{\bar{w}}+1\right)+4\\
& = &
3\ds\frac{1}{w}\ds\frac{1}{\bar{w}}+(3+10i)\ds\frac{1}{w}+(3-10i)\ds\frac{1}{\bar{w}}+7\\
& = &
\ds\frac{1}{w\bar{w}}\left(3+(3+10i)\bar{w}+(3-10i)w+7w\bar{w}\right)
\end{eqnarray*}

So, the equation for $m(A)$ is

$$\un{7w\bar{w}+(3-10i)w+(3+10i)\bar{w}+3=0}$$ which is a
euclidean circle (since the coefficient of $w\bar{w}$ is nonzero).

Determine the euclidean center and radius of $m(A)$. [10 points]

Complete the square:
\begin{eqnarray*}
& &
w\bar{w}+\ds\frac{3-10i}{7}w+\ds\frac{3+10i}{7}\bar{w}+\ds\frac{3}{7}\\
& = &
\left(w+\ds\frac{3+10i}{7}\right)\left(\bar{w}+\ds\frac{3-10i}{7}\right)
+\ds\frac{3}{7}-\ds\frac{(3+10i)(3-10i)}{49}\\ & = &
\left|w-\left(\ds\frac{-3-10i}{7}\right)\right|^2-\ds\frac{88}{49}=0
\end{eqnarray*}

So, the euclidean center is \un{$\ds\frac{-3-10i}{7}$} and the
euclidean radius is \un{$\ds\frac{\sqrt{88}}{7}$}.
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