Here we will investigate the graphs involving
and 
. We will start
with 
.
This is probably no surprise if you remember
that an equation of the form 
defines
a circle with center at the origin and radius = r.
Now, add xy to the expression, so we have 
.
Now lets vary the coefficient of the xy term,
we'll call it a, 
a=.5
a=1
a=1.5
a=1.75
a=2
a=2.5
Note that it appears that as a increases from 0 (the circle equation) to a=2, it first appears to be an ellipse then "opens up" to look like two lines around a=2. For a>2, it appears to be two hyperboli.
Now lets look at a<0.
a=-.5
a=-1
a=-1.5
a=-2
a=-2.5
For a<0, we see a similar pattern. The circle opens up to look like an ellipse, then two parallel lines at a=-2, then two hyperboli.
Click here to see an animation of this as a varies between -4 and 4.