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,
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.
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.