P-BLTZMC09_873-950-hr 21-11-2008 13:28 Page 873 From ripples in water to the path on which humanity journeys through space, certain curves occur natu- rally throughout the universe. Over two thousand years ago, the ancient Greeks studied these curves, called conic sections, without regard to their immediate usefulness simply because studying them elicited ideas that were exciting, challenging, and interesting. The ancient Greeks could not have imagined the applications of these curves in the twenty- first century. They enable the Hubble Space Telescope, a large satellite about the size of a school bus orbiting 375 miles above Earth, to gather distant rays of light and focus them into spectacular images of our evolving universe. They provide doctors with a procedure for dissolving kidney stones painlessly without invasive surgery. In this chapter, we use the rectangular coordinate system to study the conic sections and the mathematics behind their surprising applications. Here’s where you’ll find applications that move beyond planet Earth: • Planetary orbits: Section 9.1, page 881; Exercise Set 9.1, Exercise 78. • Halley’s Comet: Essay on page 882. • Hubble Space Telescope: Section 9.3, pages 900 and 908. For a kidney stone here on Earth, see Section 9.1, page 882. 873 P-BLTZMC09_873-950-hr 21-11-2008 13:28 Page 874 874 Chapter 9 Conic Sections and Analytic Geometry Section 9.1 The Ellipse Objectives ou took on a summer job driving a ᕡ Graph ellipses centered at the Ytruck, delivering books that were origin. ordered online. You’re an avid ᕢ Write equations of ellipses in reader, so just being around standard form. books sounded appealing. However, now you’re ᕣ Graph ellipses not centered at feeling a bit shaky driving the origin. the truck for the first ᕤ Solve applied problems time. It’s 10 feet wide and involving ellipses. 9 feet high; compared to your compact car, it feels like you’re behind the wheel of a tank. Up ahead you see a sign at the semielliptical entrance to a tunnel: Caution! Tunnel is 10 Feet High at Center Peak. Then you see another sign: Caution! Tunnel Is 40 Feet Wide. Will your truck clear the opening of the tunnel’s archway? Mathematics is present in the movements of planets, bridge and tunnel construction, navigational systems used to keep track of a ship’s location, manufac- ture of lenses for telescopes, and even in a procedure for disintegrating kidney stones. The mathematics behind these applications involves conic sections. Conic sections are curves that result from the intersection of a right circular cone and a plane. Figure 9.1 illustrates the four conic sections: the circle, the ellipse, the parabola, and the hyperbola. Circle Ellipse Parabola Hyperbola Pin focus 1 Figure 9.1 Obtaining the conic sections by intersecting a plane and a cone Taut In this section, we study the symmetric oval-shaped curve known as the ellipse. inflexible We will use a geometric definition for an ellipse to derive its equation. With this string equation, we will determine if your delivery truck will clear the tunnel’s entrance. Definition of an Ellipse Pin focus 2 Figure 9.2 illustrates how to draw an ellipse. Place pins at two fixed points, each of which is called a focus (plural: foci). If the ends of a fixed length of string are fastened Figure 9.2 Drawing an ellipse to the pins and we draw the string taut with a pencil, the path traced by the pencil will be an ellipse. Notice that the sum of the distances of the pencil point from the foci remains constant because the length of the string is fixed.This procedure for drawing an ellipse illustrates its geometric definition. PP Definition of an Ellipse An ellipse is the set of all points,P, in a plane the sum of whose distances from F F 1 2 two fixed points,F1 and F2 , is constant (see Figure 9.3). These two fixed points are called the foci (plural of focus). The midpoint of the segment connecting the foci is the center of the ellipse. Figure 9.3 P-BLTZMC09_873-950-hr 21-11-2008 13:28 Page 875 Section 9.1 The Ellipse 875 Figure 9.4 illustrates that an ellipse can be elongated in any direction. In this section, we will limit our discussion to ellipses that are elongated horizontally or vertically. The line through the foci intersects the ellipse at two points, called the vertices (singular: vertex). The line segment that joins the vertices is the major axis. Notice that the midpoint of the major axis is the center of the ellipse. The line segment whose endpoints are on the ellipse and that is perpendicular to the major axis at the center is called the minor axis of the ellipse. Vertex Major axis Vertex Major axis Vertex Center Center Minor axis Minor axis Vertex Figure 9.4 Horizontal and vertical elongations of an ellipse Standard Form of the Equation of an Ellipse The rectangular coordinate system gives us a unique way of describing an ellipse. It enables us to translate an ellipse’s geometric definition into an algebraic equation. y We start with Figure 9.5 to obtain an ellipse’s equation. We’ve placed an ellipse P(x, y) that is elongated horizontally into a rectangular coordinate system.The foci are on the x-axis at -c, 0 and c, 0 , as in Figure 9.5. In this way, the center of the ellipse is at d1 1 2 1 2 d2 the origin. We let x, y represent the coordinates of any point on the ellipse. x 1 2 F (−c, 0) F (c, 0) What does the definition of an ellipse tell us about the point x, y in Figure 9.5? 1 2 1 2 For any point x, y on the ellipse, the sum of the distances to the two foci, d + d , 1 2 1 2 must be constant. As we shall see, it is convenient to denote this constant by 2a. Thus, Figure 9.5 the point x, y is on the ellipse if and only if 1 2 + = Study Tip d1 d2 2a. 2 2 2 2 The algebraic details behind elimi- 4 x + c + y + 4 x - c + y = 2a Use the distance formula. 1 2 1 2 nating the radicals and obtaining the After eliminating radicals and simplifying, we obtain equation shown can be found in 2 2 2 2 2 2 2 2 Appendix A. There you will find a a - c x + a y = a a - c . step-by-step derivation of the 1 2 1 2 ellipse’s equation. Look at the triangle in Figure 9.5. Notice that the distance from F1 to F2 is 2c. Because the length of any side of a triangle is less than the sum of the lengths of the other two sides,2c 6 d1 + d2 . Equivalently,2c 6 2a and c 6 a. Consequently, a2 - c2 7 0. For convenience, let b2 = a2 - c2. Substituting b2 for a2 - c2 in the preceding equation, we obtain 2 2 2 2 2 2 b x + a y = a b 2 2 2 2 2 2 b x a y a b 2 2 + = 2 2 2 2 2 2 Divide both sides by a b . a b a b a b x2 y2 + = 1. Simplify. a2 b2 This last equation is the standard form of the equation of an ellipse centered at the origin. There are two such equations, one for a horizontal major axis and one for a vertical major axis. P-BLTZMC09_873-950-hr 21-11-2008 13:28 Page 876 876 Chapter 9 Conic Sections and Analytic Geometry Standard Forms of the Equations of an Ellipse The standard form of the equation of an ellipse with center at the origin, and major and minor axes of lengths 2a and 2b (where and ba are positive, and a2 7 b2) is x2 y2 x2 y2 + = 1 or + = 1. a2 b2 b2 a2 Study Tip Figure 9.6 illustrates that the vertices are on the major axis,a units from the center. 2 = 2 - 2 The foci are on the major axis,c units from the center. For both equations, The form c a b is the one you 2 = 2 - 2 2 = 2 - 2 should remember. When finding the b a c . Equivalently, c a b . foci, this form is easy to manipulate. y y x2 y2 x2 y2 += 1 += 1 (0, a) a2 b2 b2 a2 (0, b) (0, c) (0, 0) (0, 0) x x (−a, 0) (−c, 0) (c, 0) (a, 0) (−b, 0) (b, 0) (0, −c) (0, −b) (0, −a) Figure 9.6(a) Major axis is Figure 9.6(b) Major axis is horizontal with length 2a. vertical with length 2a. The intercepts shown in Figure 9.6(a) can be obtained algebraically. Let’s do this for x2 y2 + = 1. a2 b2 .0 ؍ y-intercepts: Set x .0 ؍ x-intercepts: Set y x2 y2 = 1 = 1 a2 b2 x2 = a2 y2 = b2 x —a y —b x-intercepts are −a and a. y-intercepts are −b and b. The graph passes through The graph passes through (−a, 0) and (a, 0), which (0, −b) and (0, b). are the vertices. Using the Standard Form of the Equation of an Ellipse We can use the standard form of an ellipse’s equation to graph the ellipse.Although the definition of the ellipse is given in terms of its foci, the foci are not part of the graph.