M408M Learning Module Pages
Main page Chapter 10: Parametric Equations and Polar CoordinatesLearning module LM 10.1: Parametrized Curves:Learning module LM 10.2: Calculus with Parametrized Curves:Learning module LM 10.3: Polar Coordinates:Learning module LM 10.4: Areas and Lengths of Polar Curves:Learning module LM 10.5: Conic Sections:Slicing a coneEllipses Hyperbolas Parabolas and directrices Completing the square Learning module LM 10.6: Conic Sections in Polar Coordinates:Chapter 12: Vectors and the Geometry of SpaceChapter 13: Vector FunctionsChapter 14: Partial DerivativesChapter 15: Multiple Integrals 
Parabolas and directricesParabolas are on the borderline between ellipses and hyperbolas. They can be approximated by ellipses with eccentricities just below 1, or by hyperbolas with eccentricity just above 1. Either way, the foci are getting farther and farther apart as the eccentricity approaches 1, and the limiting shape cannot be described by two foci. Instead, a parabola is determined by a single focus and a line called a directrix.
