Two things about conics

A real life application of conics, from xkcd.

An ellipse has the property that sounds, lights, etc. emitted at one focus, and reflected off the dish, will be reflected back to the other focus.

A few weeks ago I tried to convince my students that a parabola is just an ellipse with one of the foci at infinity (so, in particular, plotting part of a curve and saying "ooh, it looks like a parabola" isn't valid because it might be an ellipse with one of the foci really far away) but I don't think they got it. Then again, it wasn't particularly important, and we're deliberately de-emphasizing things that have to do with conic sections.

And parabolas have this same property, where "emitted at one focus" is replaced with "traveling in the same direction"; that's just an example of the point-at-infinity thing.

On a related note, someone brought this problem to me -- what is the largest n such that a regular n-sided polygon can be inscribed in a non-circular ellipse?

The answer's four. A regular polygon can always be inscribed in a circle, so the vertices of a regular polygon inscribed in an ellipse all lie on the same circle.

But the maximal number of intersections of a curve of degree m and a curve of degree n is mn, by Bezout's theorem; a curve and an ellipse are both of degree 2, so the maximal number of intersections is four.

To complete the solution, I should name an ellipse in which a square can be inscribed; take x² + 2y² = 3, which passes through (±1, ±1).