Note that in the sections above dealing with the ellipse, hyperbola, and parabola, the algebraic equations that appeared did not contain a term of the form xy. However, in our “Algebraic View of the Conic Sections,” we stated that every conic section is of the form
where A, B, C, D, E, and F are constants. In essence, all of the equations that we have studied have had B=0. So the question arises: “What role, if any, does the xy term play in conic sections? If it were present, how would that change the geometric figure?”
Well, you might also have noticed that for each of the ellipse examples we've considered, the major and minor axes were each parallel to (or coincident with) a coordinate axis; and similarly for the transverse and conjugate axes of each example hyperbola, and the directrix and axis of each parabola. Is this related to the fact that all of their equations had B=0?
Bingo!
The presence of a nonzero xy term indicates rotation of the graph of the conic section in the plane. For example, in the case of an ellipse, the major axis is no longer parallel to the x-axis or y-axis. Rather, we now have the major axis rotated by an amount that depends on A, B. and C (and which of course is zero when B=0). Let’s look at an example.
James A. Sellers