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 x

^{2}+ 2y

^{2}= 3, which passes through (±1, ±1).

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