## Nathan Wisdom

The graph of the equation is presented below. The graph is clearly a circle with radius 3 and centered at the origin.

Now on the same axes the graph of the equation

The graph is shown

Now when the coefficient of the xy term is changed with we got the following set of graphs

Click here to see the equations for the above set of graphs. You can also continue to explore the graph.

Note that there are essentially three distinct set of graphs. The circle, the ellipse, and the hyperbole. When the coefficient of xy approaches 2 the graph begin to change from an ellipse to hyperbole thus a coefficient of 2 marks a boundary between the hyperbole and the ellipse. At the boundary mark it appears as if there are two parallel lines. Look at the graph below. Is it conclusive that these lines are parallel?

To see what happen to the graph when the coefficient is sufficiently near 2 we graph the graphs with coefficients 1.9 and 2.1. The graphs are shown below.

With coefficient of 1.9 the ellipse is elongated and with a coefficient of 2.1 we got a hyperbole. Thus a boundary mark is in the interval (1.9, 2.1).

Now we try negative coefficients. When the coefficients of the xy term are negative we get the following graphs.

We get a reflection over the y axis and thus the same boundary values with a negative sign. A set of the these graphs is shown.

See the equations

Now if we combine both sets of graph on the same axes we get the following graphs.

See the equations here

Here is another look at this set of graphs, and here for the equations

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