Abstract: Any five points in the plane (no four on a line) determine a unique conic section. What can be said about a curve $C$ with the property that any five points chosen from $C$ either always determine an ellipse (or circle) or always determine a hyperbola? Such a curve is "elliptesque" or "hyperbolesque". Non-trivial examples include $y = x^3, x \ge 0$, which is hyperbolesque and $y = x^{3/2}, 1 \le x \le 1.3$, which is elliptesque. We show that if a smooth closed curve $C$ satisfies either condition, then it must be elliptesque and bound a convex region; no unbounded smooth curve can be elliptesque. Proofs are elementary.
The opening act of this talk is a discussion of the remarkable differential equation $((y'')^{-2/3})'''=0$; Sylvester observed in 1886 that the solutions to this equation are precisely the non-degenerate conic sections, simplifying a result originally proved by Monge in 1809. Two proofs of this will be given, and both are readily accessible to undergraduate math majors who have had calculus as well as linear algebra.
Departmental veterans will recognize this as a "Potpourri" talk.