The first thing that we should do is complete the square on the \(x\)’s and the \(y\)’s to see what we’ve got here. This could be a circle, ellipse, or hyperbola and completing the square a couple of times will put it into standard form and we’ll be able to identify the graph at that point.

Here is the completing the square work.

\[\begin{align*}{x^2} - 4x + \left( {4 - 4} \right) + {y^2} - 6y + \left( {9 - 9} \right) - 87 & = 0\\ {\left( {x - 2} \right)^2} + {\left( {y - 3} \right)^2} - 100 & = 0\\ {\left( {x - 2} \right)^2} + {\left( {y - 3} \right)^2} & = 100\end{align*}\]

So, we’ve got a circle with center \(\left( {2,3} \right)\) and radius 10. Here is a sketch of the circle.