Chapter 15

June 03 2017

Chapter 15 Conic Sections 15.1 Introduction: Sections of a Cone - the curves obtained by slicing a cone with a plane that does not pass through the vertex - are called conic sections, or simply conics - if the slicing plane is parallel to a generator - the conic is a parabola - otherwise - the conic is an ellipse or a hyperbola - depending on whether the plane cuts just one or both nappes - many important discoveries in both pure mathematics and science - have been linked to the conic sections - we shall be studying the conic sections as plane curves - for this purpose it is convenient to make use of equivalent definitions that refer only to the plane in which the curves lie - and depend on special points in this plane called foci - an ellipse can be defined as the set of all points in the plane the sum of whose distances from the two foci is constant - a hyperbola is the set of all points for which the difference of the distances to the foci is constant - a parabola is the set of all points for which the distance to the focus equals the distance to a fixed line (called the directrix) 15.2 Another Look at Circles and Parabolas - the reflection property of parabolas has many applications 15.3 Ellipses - we can define an ellipse as the locus of a point p - that moves so that the sum of its distances from the two foci is constant - if the minor axis is very small compared to the major axis - the ellipse is long and thin - the ratio c/a is called the eccentricity of the ellipse - e = c/a - nearly circular ellipses have e near 0 - the two foci are close to one anther - the equation of the ellipse is - x2/a2 + y2/b2 = 1 - an ellipse can be characterized as the locus of a point that moves in such a way - that the ratio of its distance from a focus to its distance to a directrix equals a constant e < 1 15.4 Hyperbolas - the equation for the hyperbola is - x2 / a2 - y2 / b2 = 1 - the ratio e = c/a is called the eccentricity of the hyperbola - e > 1 - when e is near 1 the hyperbola lies in a small angle between the asymptotes - the hyperbola can be characterized as the locus of a point that moves in such a way - that the ratio of its distance from a focus to its distance form a directrix equals a constant e > 1 15.5 The Focus-Directrix-Eccentricity Definitions - we have seen there are several distinct but equivalent ways of defining the conic sections - each with its own merits - we began with the definition by means of a given cone and a slicing plane that cuts through - yielding three types of curves - for the purpose of obtaining Cartesian equations for use in precise quantitative studies - we needed two-dimensional characterizations - for this the focal properties were convenient - the concepts of eccentricity and directrix can give yet another two-dimensional characterization by means of a focus, directrix, and eccentricity - here we show all three of the conic sections - can be given unified definitions that depend directly on our original concept of sections of a cone - by geometry - eccentricity e = cos a / cos b - this number is constant for a given cone and a given slicing plane - for e < 1, the slice is an ellipse - for e = 1, it is a parabola - for e > 1, it is a hyperbola