1010 CHAPTER 10 Analytic Geometry When a ray of light or sound emanating from one focus of an ellipse bounces off the ellipse, it passes through the other focus. See Figure 26. This reflecting property is responsible for “whispering galleries.” In a whispering gallery, a person whispering at a certain point in the room can be heard clearly at another point across the room. The U.S. statesman John Quincy Adams was able to listen in on his opponents’ conversations in the old House Chamber (Statuary Hall) because his desk was positioned at one of the foci beneath the ellipsoidal ceiling and his opponents were located across the room at the other focus. A lithotripter is a machine that uses shock waves to crush kidney stones. The patient is placed in an elliptical tub with the kidney stone at one focus of the ellipse. A beam is projected from the other focus to the tub so that it reflects to hit the kidney stone. See Figure 27. x y Aerial view of Old House Chamber Figure 26 Kidney stone Emitter Focusing reflector Shock waves The top of an ellipse is illustrated in this depiction of how a lithotripter crushes a kidney stone. Figure 27 10.2 Exercises 1. CONCEPT PREVIEW Match each equation of an ellipse in Column I with the appropriate intercepts in Column II. I (a) 36x2 + 9y2 = 324 (b) 9x2 + 36y2 = 324 (c) x2 25 + y2 16 = 1 (d) x2 16 + y2 25 = 1 II A. 1-3, 02, 13, 02, 10, -62, 10, 62 B. 1-4, 02, 14, 02, 10, -52, 10, 52 C. 1-6, 02, 16, 02, 10, -32, 10, 32 D. 1-5, 02, 15, 02, 10, -42, 10, 42 EXAMPLE 8 Modeling the Reflective Property of Ellipses If a lithotripter is based on the ellipse x2 36 + y2 27 = 1, determine how many units both the kidney stone and the source of the beam must be placed from the center of the ellipse. SOLUTION The kidney stone and the source of the beam must be placed at the foci, 1c, 02 and 1-c, 02. Here a2 = 36 and b2 = 27. c2 = a2 - b2 Relationship for ellipses c2 = 36 - 27 Let a2 = 36 and b2 = 27. c2 = 9 Subtract. c = 3 Take the positive square root. Thus, the foci are 13, 02 and 1-3, 02. The kidney stone and the source both must be placed on the major axis 3 units from the center. See Figure 28. S Now Try Exercise 61. (3, 0) (–3, 0) Kidney stone Wave source Wave path y 6 –6 x + = 1 36 x2 27 y2 –Ë27 Ë27 Figure 28
RkJQdWJsaXNoZXIy NjM5ODQ=