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\displaystyle[br]\begin{equation*} \frac{z-3i}{z-5} = \frac{x + (y-3)i}{(x-5) + iy} = \frac{(x +(y-3)i)((x-5) -iy) }{(x-5)^2 + y^2}
\displaystyle [br]\begin{equation*}\frac{x(x-5) + y(y+3) - iyx + (y-3)(x-5)i}{(x-5)^2 + y^2} \end{equation*}
\displaystyle[br]\begin{equation*}x^2 - 5x + y^2 + 3y = -5y - 3x + 15\end{equation*}
\displaystyle[br]\begin{equation*}(x-1)^2 + (y+1)^2 = 15 + 2 = 17\end{equation*}
\displaystyle[br]\begin{equation*} \frac{z-3i}{z-5} = \frac{x + (y-3)i}{(x-5) + iy} = \frac{(x +(y-3)i)((x-5) -iy) }{(x-5)^2 + y^2}
\displaystyle [br]\begin{equation*}\frac{x(x-5) + y(y+3) - iyx + (y-3)(x-5)i}{(x-5)^2 + y^2} \end{equation*}
\displaystyle[br]\begin{equation*}x^2 - 5x + y^2 + 3y = -5y - 3x + 15\end{equation*}
\displaystyle[br]\begin{equation*}(x-1)^2 + (y+1)^2 = 15 + 2 = 17\end{equation*}
Spoiler
\displaystyle[br]\begin{equation*} \frac{z-3i}{z-5} = \frac{x + (y-3)i}{(x-5) + iy} = \frac{(x +(y-3)i)((x-5) -iy) }{(x-5)^2 + y^2}
\displaystyle [br]\begin{equation*}\frac{x(x-5) + y(y+3) - iyx + (y-3)(x-5)i}{(x-5)^2 + y^2} \end{equation*}
\displaystyle[br]\begin{equation*}x^2 - 5x + y^2 + 3y = -5y - 3x + 15\end{equation*}
\displaystyle[br]\begin{equation*}(x-1)^2 + (y+1)^2 = 15 + 2 = 17\end{equation*}
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