Sketch Conics: Motivating Students to Learn Cross Products

[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?

 

[Vargo]

Understanding the second derivative test for multivariable functions could be a motivating factor. It is not exactly the same thing as sketching conics, but it is closely related.