College Algebra, Chapter 2, Review Exercises, Section Review Exercises, Problem 14

Use completing the square to determine whether the equation $x^2 + y^2 - 6x - 10y + 34 = 0$ represents a circle or a point or has no graph.

$
\begin{equation}
\begin{aligned}
x^2 + y^2 - 6x - 10y + 34 &= 0 && \text{Model}\\
\\
\left( x^2 - 6x + \underline{\phantom{xx }} \right) + \left( y^2 - 10y + \underline{\phantom{xx }} \right) &= -34 && \text{Group terms}\\
\\
\left( x^2 - 6x + 9 \right)^3 + ( y^2 - 10y + 25 ) &= -34 + 25 + 9 && \text{Complete the square: Add } \left( -\frac{6}{2} \right)^2 = 9 \text{ and } \left( \frac{-10}{2} \right)^2 = 25 \\
\\
(x-3)^2 + (y-5)^2 &= 0 && \text{Perfect square, the equation represents a point.}
\end{aligned}
\end{equation}
$