MHB Sketch Conics: Motivating Students to Learn Cross Product Terms

[matqkks]

What is the most motivating way to introduce the sketching of conics which have a cross product terms?
This topic involves a lot of other stuff such as eigenvalues, orthogonal matrices, completing the square etc. I find a significant number of students get lost in this forest of sketching conics. Are there examples which have a real impact and are motivating why they should learn this topic?

 

[Fernando Revilla]

Perhaps a motivating way is to comment the standard sketching: $$\begin{aligned}ax^2+by^2+cxy+dx+ey+f=0& \Leftrightarrow by^2+(cx+e)y+ax^2+dx+f=0\\ &\Leftrightarrow y=\dfrac{-(cx+e)\pm\sqrt{\Delta}}{2b}\end{aligned}$$ where $\Delta =(cx+e)^2-4b(ax^2+dx+f)$. Now we have to factorize $\Delta$ for finding the domain, obtaining two branches $y=g(x)+\sqrt{\Delta}$ and $y=g(x)-\sqrt{\Delta}$. Choose an example so that the students can compare with algebraic methods. At least, we gain in elegance.

 

[Ackbach]

All solutions of the two-body problem with an inverse-square force law (so, two bodies with the gravitational force between them, or the electrical force) are conic sections. If the total energy is low, the orbits are circles. Then, as the energy increases, you change the angle of the section you're taking - circle to ellipse to parabola to hyperbola. Much of the time, you can choose your coordinate system so that there are no cross terms, but not always.

To summarize: conic sections describe planet orbits, as well as the motion of two oppositely charged particles exerting an attractive electrical force on each other.