**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