How to Recognize a Parabola

Parabolas have many interesting properties which were perhaps more well known in centuries past. Many of these properties hold only for parabolas, providing a characterization which can be used to recognize (theoretically, at least) a parabola. Here, we present a dozen characterizations of parabolas, involving tangent lines, areas, and the well-known reflective property. While some of these properties are widely known to hold for parabolas, the fact that they hold only for parabolas may be less well known. These remarkable properties can be verified using only elementary techniques of calculus, geometry, and differential equations. A parabola is the set of points in the plane which are equidistant from a point F called the focus and a line l called the directrix. If the directrix is horizontal, then the parabola is the graph of a quadratic function p(x) = αx2 +βx+∞. We will not distinguish between a function and its graph. A chord of a function f(x) is a line segment whose endpoints lie on the graph of the function. A chord of f(x) is a segment of a secant line to f(x). By the equation of a chord, we mean the equation of the corresponding secant line. Our first characterization of parabolas involves the area between a function and a chord having horizontal extent h.