**Exploring
the graph of x ^{2}+nxy+y^{2}=9**

Consider the graph of the equation x^{2}+y^{2}=9.

This is the graph of a circle centered at the
origin with radius equal to 3. The equation x^{2}+y^{2}=9 can
be thought of as a special case of the more general equation x^{2}+nxy+y^{2}=9
where n=0. Consider now some graphs of the general equation for different
values of n.

We get a series of ellipses (pointing in the
direction of the II and IV quadrants) and hyperbolas (opening up towards the I
and III quadrants). We note that all curves pass through the same 4 points:
(3,0), (0,3), (-3,0), and (0,-3). The graph when n=2 suggests that this may be
a critical value where the graph transitions from one kind of curve to another
kind.

Note that, so far, we have only looked at
positive values for n. Before we go further, let's look at the case where n
takes on negative values.

We see that we get the same curves simply
reflected across the x-axis (or the y-axis). Now the ellipses point in the
direction of the I and III quadrants and the hyperbolas open up in the
direction of the II and IV quadrants. So we can limit our exploration to
positive values of n and know that we can extrapolate our results by reflecting
over either axis.

Now lets get back to looking at what happens
when n=2. First, let's examine the curves when n varies from 0 to 2.

So starting with our original circle when
n=0, we see that the curves become an increasingly stretched ellipse until n
reaches 2. The graph when n=2 looks like two parallel lines with the scale set
as it is. In fact, it is still an ellipse that has just been stretched out to
infinity.

We will now look at the curves when n is
greater than or equal to 2.

So we see that as n increases, for n>2,
that we get hyperbolas with a sharper vertex that gets closer and closer to the
origin.