680 Analytic Geometry A Look Back In Chapter 1, we introduced rectangular coordinates and showed how geometry problems can be solved algebraically. We defined a circle geometrically and then used the distance formula and rectangular coordinates to obtain an equation for a circle. A Look Ahead In this chapter, geometric definitions are given for the conics , and the distance formula and rectangular coordinates are used to obtain their equations. Conics, and their unique properties, are frequently used in a wide variety of applications. Paraboloids of revolution (parabolas rotated about their axes of symmetry) are used as signal collectors (the satellite dishes used with radar and dish TV, for example), as solar energy collectors, and as reflectors (telescopes, light projection). The planets circle the Sun in approximately elliptical orbits. Elliptical surfaces are used to reflect signals such as light and sound from one place to another. A third conic, the hyperbola , is used to determine the location of ships or sound sources, such as lightning strikes. The Greeks used Euclidean geometry to study conics. However, we use the more powerful methods of analytic geometry, which uses both algebra and geometry, for our study of conics. In Section 10.7, we introduce parametric equations , which allow us to represent graphs of curves that are not the graph of a function, such as a circle. The Orbit of Comet Hale-Bopp The orbits of Comet Hale-Bopp and Earth can be modeled using ellipses , the subject of Section 10.3. The Internet-based Project at the end of this chapter explores the possibility of Comet Hale-Bopp colliding with Earth. —See the Internet-based Chapter Project— Outline 10. 1 Conics 10. 2 The Parabola 10. 3 The Ellipse 10. 4 The Hyperbola 10. 5 Rotation of Axes; General Form of a Conic 10. 6 Polar Equations of Conics 10. 7 Plane Curves and Parametric Equations Chapter Review Chapter Test Cumulative Review Chapter Project 10 Credit: MarcelClemens/Shutterstock
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