A CHARACTERIZATION OF CONIC SECTIONS

Abstract

Abstract. We study some properties of tangent lines of conic sec-tions. As a result, we establish a characterization of conic sections. 1. IntroductionThe best known plane curves are straight lines and circles, which arecharacterized as the plane curves of constant curvature. The next mostfamiliar plane curves are arguably the conic sections.A di erential geometric characterization of conic sections were stud-ied by the rst author and others in terms of the curvature and thesupport function ([3], [8]).As was described in [5], a circle is characterized by the fact that thechord joining any two points on it meets the circle at the same angle.For a generalization of this property to space curves, see [1]. In thisregard, it is interesting to consider what simple geometric propertiescharacterize conic sections.In this note, we examine the conic sections concerning the chordpassing through a focus and discuss the converse problems of well knownproperties about the conic sections.Let’s x a line Land a point F which is not on L. For a positiveconstant e, consider the locus C(e) of points Pwhose distances from Fand Lare, respectively, in a constant ratio e. Then the locus C(e) is aconic section with a focus F, a corresponding directrix Land eccentricitye. That is, C(e) is a parabola if e= 1; an ellipse if 0 1.Then the following are well-known, or can be easily checked.