Carrie Dillman

The xy Term

Consider the equation x^2+y^2=9 and its graph.

Now consider the equation x^2+xy+y^2=9 and its graph.

The xy term "stretches" the circle resulting in an ellipse. Did we just add the xy term? No, the xy term is actually in the equation of the circle as well. Compare the coefficients of the xy term in each equation. Clearly, the coefficient of the ellipse is an unstated 1. Therefore, the coefficient of the "unseen" xy term in the equation of the circle must be zero.

Let's build on the above graph, trying different coefficients and see what we come up with.

Interesting! The values of the coefficients used above are 0 through 5, 10 and 20. Recall that the red ellipse is the result of the coefficient 1. What are those blue parallel lines all about though? In order to get the whole story we need to see what effect xy coefficients between 1 and 2 have on the graph.

So alot is happening between the coefficients 1 and 2. We see that the ellipse is stretching until it "breaks" and splits into the two parallel lines.

Wait, wait, wait! Does the same thing happen when we use negative numbers? Let's see.

Wow it does! In both cases, the circle is "stretched" into an ellipse and the ellipse is "stretched" even further until it comes to a point where it breaks apart and forms a set of parallel lines and those lines curve forming hyperbolas.

Now let's overlay the graphs of the positive coefficients with the graphs of the negative coefficients.

Ooohh! How pretty. How can you not love mathematics.

Let's recap. We have managed to generate four different graphs, a circle, an ellipse, parallel lines, and a hyperbola, from this one equation, x^2+y^2=9. What coefficients mark the boundaries between these graphs? Obviously, the coefficient 0 is a boundary because x^2+0xy+y^2=9 is a perfect circle. The coefficient 1 is not a boundary because we can infer from the above graphs that coefficients between 0 and 2 create different elliptical curves. The other boundaries are the coefficients 2 and -2, at which points the graph is a set of parallel lines. Coefficients greater than 2 or less than -2 form hyperbola curves.