688 CHAPTER 10 Analytic Geometry 3 Solve Applied Problems Involving Parabolas Parabolas occur in many applications. For example, as discussed in Section 3.4, suspension bridges have cables in the shape of a parabola. Parabolas also have a reflecting property that is used in applications. A parabola that is rotated about its axis of symmetry generates a surface called a paraboloid of revolution. If a light (or any other emitting source) is placed at the focus of the parabola, all the rays emanating from the light will reflect off the paraboloid of revolution in lines parallel to the axis of symmetry of the parabola. This principle is used in the design of searchlights, flashlights, certain automobile headlights, and other such devices. See Figure 16. Conversely, suppose that rays of light (or other signals) emanate from a distant source so that they are essentially parallel. When these rays strike the surface of a parabolic mirror whose axis of symmetry is parallel to these rays, they are reflected to a single point at the focus. This principle is used in the design of some solar energy devices, satellite dishes, and the mirrors used in some types of telescopes. See Figure 17. Satellite Dish A satellite dish is shaped like a paraboloid of revolution. The signals that emanate from a satellite strike the surface of the dish and are reflected to a single point, where the receiver is located. If the dish is 8 feet across at its opening and 3 feet deep at its center, at what position should the receiver be placed? That is, where is the focus? Solution EXAMPLE 10 Figure 18(a) shows the satellite dish. On a rectangular coordinate system, draw the parabola used to form the dish so that the vertex of the parabola is at the origin and its focus is on the positive y-axis. See Figure 18(b). Figure 18 (b) Satellite dish model 8' (a) USA Cable 3' x y 4 22 2 24 0 2 4 (4, 3) (24, 3) 8' 3' F 5 (0, a) Figure 16 Searchlight Light at focus Rays of light Figure 17 Telescope The form of the equation of the parabola is = x ay 4 2
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