SECTION 10.1 Conics 681 Figure 1 Right circular cone Vertex, V Nappes g Generators Axis, a 10.1 Conics OBJECTIVE 1 Know the Names of the Conics (p. 681) 1 Know the Names of the Conics The word conic derives from the word cone , which is a geometric figure that can be constructed in the following way: Let a and g be two distinct lines that intersect at a point V. Keep the line a fixed. Now rotate the line g about a, while maintaining the same angle between a and g. The collection of points swept out (generated) by the line g is called a right circular cone . See Figure 1. The fixed line a is called the axis of the cone; the point V is its vertex ; the lines that pass through V and make the same angle with a as g are generators of the cone. Each generator is a line that lies entirely on the cone. The cone consists of two parts, called nappes , that intersect at the vertex. Conics , an abbreviation for conic sections , are curves that result from the intersection of a right circular cone and a plane.We discuss only conics formed where the plane does not contain the vertex. These conics are circles when the plane is perpendicular to the axis of the cone and intersects each generator; ellipses when the plane is tilted slightly so that it intersects each generator, but intersects only one nappe of the cone; parabolas when the plane is tilted farther so that it is parallel to one (and only one) generator and intersects only one nappe of the cone; and hyperbolas when the plane intersects both nappes. See Figure 2. Figure 2 (a) Circle Axis (b) Ellipse Axis (c) Parabola Axis Generator (d) Hyperbola Axis If the plane contains the vertex, the intersection of the plane and the cone is a point, a line, or a pair of intersecting lines. These are called degenerate conics . Conic sections are used in modeling many applications. For example, parabolas are used in describing searchlights and telescopes (see Figures 16 and 17 on page 688). Ellipses are used to model the orbits of planets and whispering chambers (see page 700).And hyperbolas are used to locate lightning strikes and model nuclear cooling towers (see Problems 78 and 79 in Section 10.4). Ada Lovelace (1815–1852) English mathematician Ada Lovelace is considered to be the first computer programmer. Ada realized that a computer could follow a series of simple instructions, a program, to perform a complex calculation. The programming language called Ada, that was created in the early 1980s, was named for her. The second Tuesday in October has become Ada Lovelace Day, on which the contributions of women to science, technology, engineering, and mathematics are honored. Credit: GL Archive/Alamy Stock Photo
RkJQdWJsaXNoZXIy NjM5ODQ=