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Section 1.10 : Common Graphs

14. Without using a graphing calculator sketch the graph of \({x^2} - 4x + {y^2} - 6y - 87 = 0\).

Hint : Complete the square a couple of times to put this into standard from. This will allow you to identify the type of graph this will be.
Show Solution

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.