Usefulness of Conic Sections Conic sections are a frequent target for dropping when attempting to make room for other topics in advanced algebra and precalculus courses.  A common argument in favor of dropping them is that typical first-year calculus doesn't use conic sections at all.  Do conic sections come up in typical intro-level undergraduate courses?  In typical prelim grad-level courses?  If so, where?

If you're a physics sort of person, conic sections clearly come up when you study how Kepler figured out what the shapes of orbits are, and some of their synthetic properties give useful shortcuts to things like proving "equal area swept out in equal time" that need not involve calculus.
The other skills you typically learn while studying conic sections in analytic geometry - polar parametrization of curves, basic facts about various invariants related to triangles and conics, rotations and changing coordinate systems (so as to recognize the equation of a conic in general form as some sort of transformation of a standard done), are all extremely useful in physics. I'd say that plane analytic geometry was the single most useful math tool for me in solving physics problems until I got to fluid dynamics stuff (where that is replaced by complex analysis).
Relatedly, independent of their use in physics, I think they're a great way to show the connections between analytic and synthetic thinking in math, which will come up over and over again for people who go on to study math (coordinate-based versus intrinsic perspectives, respectively).
