College Algebra, Chapter 2, 2.2, Section 2.2, Problem 76

Sketch the graph of the equation $x^2 + y^2 - 16x + 12y + 200 = 0$


$
\begin{equation}
\begin{aligned}

x^2 + y^2 - 16x + 12y + 200 =& 0
&& \text{Model}
\\
\\
(x^2 - 16x) + (y^2 + 12y) =& -200
&& \text{Group terms and subtract } 200
\\
\\
(x^2 - 16x + 64) + (y^2 + 12y + 36) =& -200 + 64 + 36
&& \text{Complete the square: add } \left( \frac{-16}{2} \right)^2 = 64 \text{ and } \left( \frac{12}{2} \right)^2 = 36
\\
\\
(x - 8)^2 + (y + 6)^2 =& -100
&& \text{Perfect Square}

\end{aligned}
\end{equation}
$


Recall that the general equation for the circle with
circle $(h,k)$ and radius $r$ is..

$(x - h)^2 + (y - k)^2 = r^2$

By observation,

The center is at $(8,-6)$ but the radius is not a real number, in fact, it is $10i$.