-,- 2..J. EXAMPLE 9 Discussing the Equation of a Parabola. Solution

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1 670 CHAPTER 9 Analtic Geometr Polnomial equations define parabolas whenever the involve two variables that are quadratic in one variable and linear in the other. To discuss this tpe of equation, we first complete the square of the variable that is quadratic. EXAMPLE 9 Discussing the Equation of a Parabola Discuss the equation: X2 + 4x - 4 = 0 Figure 15 X2 + 4x - 4 = 0 Axis of smmetr x= -2 4 _3D:=-2 Solution 4 x To discuss the equation X2 + 4x - 4Y = 0, we complete the square involving the variable x. X2 + 4x - 4 = 0 X2 + 4x = 4 Isolate the terms involving x on the left side. X2 + 4x + 4 = Complete the square on the left side. (x + 2)2 = 4( + 1) Factor. This equation is of the form (x - h? = 4a( - k), with h = -2, k = -1, and a = 1. The graph is a parabola with vertex at (h, k) = (-2, -1) that opens up. The focus is at (-2,0), and the directrix is the line = -2. See Figure 15. QrnC=::;;;>-- NOW WORK PROBLEM J Parabolas find their wa into man applications. For example, as we discussed in Section 3.1, suspension bridges have cables in the shape of a parabola. Another propert of parabolas that is used in applications is their reflecting propert. Reflecting Propert Suppose that a mirror is shaped like a paraboloid of revolution, a surface formed b rotating a parabola about its axis of smmetr. If a light (or an other emitting source) is placed at the focus of the parabola, all the ras emanating from the light will reflect off the mirror in lines parallel to the axis of smmetr. This principle is used in the design of searchlights, flashlights, certain automobile headlights, and other such devices. See Figure 16. Conversel, suppose that ras of light (or other signals) emanate from a distant source so that the are essentiall parallel. When these ras strike the surface of a parabolic mirror whose axis of smmetr is parallel to these ras, the are reflected to a single point at the focus. This principle is used in the design of some solar energ devices, satellite dishes, and the mirrors used in some tpes of telescopes. See Figure 17. Figure 16 Searchlight Figure 17 Telescope ~ Light at --~~ focus I L-~ -,- i,> I _-- -~-_~

2 SECTION 9.2 The Parabola 671 EXAMPLE 10 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 is 3 feet deep at its center, at what position should the receiver be placed? Sol uti 0 n Figure 18(a) shows the satellite dish. We draw the parabola used to form the dish on a rectangular coordinate sstem so that the vertex of the parabola is at the origin and its focus is on the positive -axis. See Figure 18(b). Figure ' (-4,3) 4 (4,3) 3 2 F=(O,a) x (a) (b) The form of the equation of the parabola is X2 = 4a and its focus is at (0, a). Since (4,3) is a point on the graph, we have 4 2 = 4a(3) 4 a=- 3 ~:~:~~~.ShOUld be located 1~ feet from the base of the dish, along its a= Cl!l!=====>'- NOW WORK PROBLEM Concepts and Vocabular In Problems 1-3, Jill in the blanks. 1. A(n) is the collection of all points in the plane such that the distance from each point to a fixed point equals its distance to a fixed line. 2. The surface formed b rotating a parabola about its axis of smmetr is called a _ 3. The line segment joining the two points on a parabola above and below its focus is called the _ In Problems 4-6, answer True or False to each statement. 4. The vertex of a parabola is a point on the parabola that also is on its axis of smmetr. 5. If a light is placed at the focus of a parabola, all the ras reflected off the parabola will be parallel to the axis of smmetr. 6. The graph of a quadratic function is a parabola.

3 SECTION 9.2 The Parabola Focus at (0, - 3); vertex at (0,0) 19. Focus at (-2,0); directrix the line x = Directrix the line = -~; vertex at (0,0) '23. Vertex at (2, -3); focus at (2, -5) 25. Vertex at (0,0); axis of smmetr the -axis; containing the point (2,3) 27. Focus at (- 3,4); directrix the line = Focus at (-3, -2); directrix the line x = Focus at (-4,0); vertex at (0,0) 20. Focus at (0, -1); directrix the line = Directrix the line x = -~; vertex at (0,0) 24. Vertex at (4,-2); focus at (6, -2) 26. Vertex at (0,0); axis of smmetr the x-axis; containing the point (2,3) 28. Focus at (2,4); directrix the line x = Focus at (-4,4); directrix the line = -2 In Problems 31-48, find the vertex, focus, and directrix of each parabola. Graph the equation (a) b hand and (b) b using a graphing utilit. 31. x2 = = 8x 33. i = -16x 34. X2 = ( - 2)2 = 8(x + 1) 36. (x + 4)2 = 16( + 2) 37. (x - 3)2 = -( + L) 38. ( + 1? = -4(x - 2) 39. ( + 3)2 = 8(x - 2) 40. (x - 2)2 = 4( - 3) 41. i x + 4 = X2 + 6x - 4Y + 1 = X2 + 8x = 4Y i -2Y = 8x i x = X2-4x = x2-4x = Y = -x + 1 In Problems 49-56, write an equation for each parabola (0,1) -2 x x -2 x (0, -1) Y (2,2) (0,1) (0,1) -2 2 x -2 x 2 x -2 x -2-2 (1, -1) 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 10 feet across at its opening and is 4 feet deep at its center, at what position should the receiver be placed? 58. Constructing a TV Dish A cable TV receiving dish is in the shape of a paraboloid of revolution. Find the location of the receiver, which is placed at the focus, if the dish is 6 feet across at its opening and 2 feet deep. 59. Constructing a Flashlight The reflector of a flashlight is in the shape of a paraboloid of revolution. Its diameter is 4 inches and its depth is 1 inch. How far from the vertex should the light bulb be placed so that the ras will be reflected parallel to the axis?

4 674 CHAPTER 9 Analtic Geometr 60. Constructing a Headlight A sealed-beam headlight is in the shape of a paraboloid of revolution. The bulb, which is placed at the focus, is 1 inch from the vertex. If the depth is to be 2 inches, what is the diameter of the headlight at its opening? 61. Suspension Bridge The cables of a suspension bridge are in the shape of a parabola, as shown in the figure. The towers supporting the cable are 600 feet apart and 80 feet high. If the cables touch the road surface midwa between the towers, what is the height of the cable at a point 150 feet from the center of the bridge? 66. Reflecting Telescope A reflecting telescope contains a mirror shaped like a paraboloid of revolution. If the mirror is 4 inches across at its opening and is 3 feet deep, where will the collected light be concentrated? 67. Parabolic Arch Bridge A bridge is built in the shape of a parabolic arch. The bridge has a span of 120 feet and a maximum height of 25 feet. See the illustration. Choose a suitable rectangular coordinate sstem and find the height of the arch at distances of 10, 30, and 50 feet from the center. 62. Suspension Bridge The cables of a suspension bridge are in the shape of a parabola. The towers supporting the cable are 400 feet apart and 100 feet high. If the cables are at a height of 10 feet midwa between the towers, what is the height of the cable at a point 50 feet from the center of the bridge? 63. Searchlight A searchlight is shaped like a paraboloid of revolution. If the light source is located 2 feet from the base along the axis of smmetr and the opening is 5 feet across, now deep should the searchlight be? 64. Searchlight A searchlight is shaped like a paraboloid of revolution. If the light source is located 2 feet from the base along the axis of smmetr and the depth of the searchlight is 4 feet, what should the width of the opening be? 65. Solar Heat A mirror is shaped like a paraboloid of revolution and will be used to concentrate the ras of the sun at its focus, creating a heat source. (See the figure.) If the mirror is 20 feet across at its opening and is 6 feet deep, where will the heat source be concentrated? Sun's ras. 68. Parabolic Arch Bridge A bridge is to be built in the shape of a parabolic arch and is to have a span of 100 feet. The height of the arch a distance of 40 feet from the center is to be 10 feet. Find the height of the arch at its center. 69. Show that an equation of the form AX2 + E = 0, A "* 0, E "* 0 is the equation of a parabola with vertex at (0,0) and axis of smmetr the -axis, Find its focus and directrix. 70. Show that an equation of the form C2+Dx=O, C"*O,D"*O is the equation of a parabola with vertex at (0,0) and axis of smmetr the x-axis. Find its focus and directrix. 71. Show that the graph of an equation of the form AX2 + Dx + E + F = 0, A"* 0 (a) Is a parabola if E "* O. (b) Is a vertical line if E = 0 and D 2-4AF = O. (c) Is two vertical lines if E = 0 and D 2-4AF > O. (d) Contains no points if E = 0 and D 2-4AF < O. 72. Show that the graph of an equation of the form C2 + Dx + E + F = 0, C"* 0 (a) Is a parabola if D "* O. (b) Is a horizontal line if D = 0 and E 2-4CF = O. (c) Is two horizontallines if D = 0 and E 2-4CF > O. (d) Contains no points if D = 0 and E 2-4CF < O. PREPARING FOR THIS SECTION Before getting started, review the following:./ Distance Formula (Section 1.1, p. 5)./ Smmetr (Section 1.3, pp )./ Completing the Square (Section A.6, pp )./ Intercepts (Section 1.2, pp )./ Square Root Method (Section A.6, )./ Circles (Section 1.3, pp )./ Graphing Techniques: Transformations (Section 2.5, pp )

5 1-001 o(f SECTION 9.3 The Ellipse 683 Applications 5 Ellipses are found in man applications in science and engineering. For example, the orbits of the planets around the Sun are elliptical, with the Sun's position at a focus. See Figure 32. Figure 32 Stone and concrete bridges are often shaped as semielliptical arches. Elliptical gears are used in machiner when a variable rate of motion is required. Ellipses also have an interesting reflection propert. If a source of light (or sound) is placed at one focus, the waves transmitted b the source will reflect off the ellipse and concentrate at the other focus.this is the principle behind whispering galleries, which are rooms designed with elliptical ceilings. A person standing at one focus of the ellipse can whisper and be heard b a person standing at the other focus, because all the sound waves that reach the ceiling are reflected to the other person. ~XAMPLE 9 Whispering Galleries Sol uti 0 n Figure 33 shows the specifications for an elliptical ceiling in a hall designed to be a whispering galler. In a whispering galler, a person standing at one focus of the ellipse can whisper and be heard b another person standing at the other focus, because all the sound waves that reach the ceiling from one focus are reflected to the other focus. Where are the foci located in the hall? We set up a rectangular coordinate sstem so that the center of the ellipse is at the origin and the major axis is along the x-axis. See Figure 34.The equation of the ellipse is where a = 25 and b = 20. Figure 33 Figure ' " 50'----~ T 20' ( 50' -I ~ +.i. = 1 a = 25,b = 20 a 2 b 2 t 30 x

6 686 CHAPTER 9 Analtic Geometr In Problems 55-58, graph each function b hand. Use a graphing utilit to verif our results. [Hint: Notice that each function is half an ellipse.] 55. f(x) = V16-4X2 56. f(x) = V9-9x f(x) = - V64-16x f(x) = -V4-4x Semielliptical Arch Bridge An arch in the shape of the upper half of an ellipse is used to support a bridge that is to span a river 20 meters wide. The center of the arch is 6 meters above the center ofthe river (see the figure). Write an equation for the ellipse in which the x-axis coincides with the water level and the -axis passes through the center of the arch. 60. Semi elliptical Arch Bridge The arch of a bridge is a semiellipse with a horizontal major axis. The span is 30 feet, and the top of the arch is 10 feet above the major axis.the roadwa is horizontal and is 2 feet above the top of the arch. Find the vertical distance from the roadwa to the arch <ilt5-footintervals along the roadwa. 61. Whispering Galler A hall 100 feet in length is to be designed asa whispering galler. If the foci are located 25 feet from the center, how high will the ceiling be at the center? 62. Whispering Galler Jim, standing at one focus of a whispering galler, is 6 feet from the nearest wall. His friend is standing at the other focus, 100 feet awa. What is the length of this whispering galler? How high is its elliptical ceiling at the center? 63. SemiellipticaI Arch Bridge A bridge is built in the shape of a semielliptical arch. The bridge has a span of 120 feet and a maximum height of 25 feet. Choose a suitable rectangular coordinate sstem and find the height of the arch at distances of 10, 30, and 50 feet from the center. 64. Semielliptical Arch Bridge A bridge is to be built in the shape of a semielliptical arch and is to have a span of 100 feet. The height of the arch, at a distance of 40 feet from the center, is to be 10 feet. Find the height of the arch at its center. 65. SemiellipticaI Arch An arch in the form of half an ellipse is 40 feet wide and 15 feet high at the center. Find the height of the arch at intervals of 10 feet along its width. 66. Semielliptical Arch Bridge An arch for a bridge over a highwa is in the form of half an ellipse. The top of the arch is 20 feet above the ground level (the major axis). The highwa has four lanes, each 12 feet wide; a center safet strip 8 feet wide; and two side strips, each 4 feet wide. What should the span of the bridge be (the length of its major axis) if the height 28 feet from the center is to be 13 feet? In Problems.67-70, use the fact that the orbit of a planet about the Sun is an ellipse, with the Sun at one focus. The aphelion of a planet is its greatest distance from the Sun, and the perihelion is its shortest distance. The mean distance of a planet from the Sun is the length of the semimajor axis of the elliptical orbit. See the illustration. 67. Earth The mean distance of Earth from the Sun is 93 million miles. If the aphelion of Earth is 94.5 million miles, what is the perihelion? Write an equation for the orbit of Earth around the Sun. 68. Mars The mean distance of Mars from the Sun is 142 million miles. If the perihelion of Mars is million miles, what is the aphelion? Write an equation for the orbit of Mars about the Sun. 69. Jupiter The aphelion of Jupiter is 507 million miles. If the distance from the Sun to the center of its elliptical orbit is 23.2 million miles, what is the perihelion? What is the mean distance? Write an equation for the orbit of. Jupiter around the Sun. 70. Pluto The perihelion of Pluto is 4551 million miles, and the distance of the Sun from the center of its elliptical orbit is million miles. Find the aphelion of Pluto. What is the mean distance of Pluto from the tii? Write an equation for the orbit of Pluto about the Sun.

7 SECTION 9.4 The Hperbola Racetrack Design Consult the figure. A racetrack is in the shape of an ellipse, 100 feet long and 50 feet wide. What is the width 10 feet from a vertex? 74. Show that the graph of an equation of the form AX2 + ci + Dx + E + F = 0, where A and C are of the same sign, A * O,C * 72. Racetrack Design A racetrack is in the shape of an ellipse 80 feet long and 40 feet wide. What is the width 10 feet from a vertex? ~ 75. The_eccentricite of an ellipse is defined as the number 73. Show that an equation of the form.., where a and c are the numbers given in equation (2). AX2 + c2 + F = 0, A * 0, C * 0, F * a- where A and C are of the same sign and F is of opposite sign, (a) Is the equation of an ellipse with center at (0,0) if A * C. (b) Is the equation of a circle with center (0,0) if A = C. ( ) I 11 f D2 E 2 F h. a s an e Ipse I 4A + 4C - ISt e same sign as A. D 2 E 2 (b) Is a point if 4A + 4C - F = 0.. D 2 E 2 (c) ~;:~~~. no points if 4A + 4C - F is of opposite Because a > c, it follows that e < 1. Write a brief paragraph about the general shape of each of the following ellipses. Be sure to justif our conclusions. (a) Eccentricit close to (b) Eccentricit = 0.5 (c) Eccentricit close to 1 PREPARING FOR THIS SECTION Before getting started, review the following:.i Distance Formula (Section 1.1, p. 5).I Completing the Square (Section A.6, pp ).I Smmetr (Section 1.3, pp ).I Asmptotes (Section 3.4, pp ).I Graphing Techniques: Transformations (Section 2.5, pp ).I Square Root Method (Section A.6, pp ) IEIITHE HYPERBOLA OBJECTIVES»Find the Equation of a Hperbola.v Graph Hperbolas 2..J Discuss the Equation of a Hperbola 4) Find the Asmptotes of a Hperbola V Work with Hperbolas with Center at (h, k) 6 Solve Applied Problems Involving Hperbolas A hperbola is the collection of all points in the plane the difference of whose distances from two fixed points, called the foci, is a constant..!.j Figure 35 illustrates a hperbola with foci FI and F 2. The line containing the foci is called the transverse axis. The midpoint of the line segment joining the foci is called the center of the hperbola. The line through the center

8 SECTION 9.4 The Hperbola 697 This is the equation of a hperbola with center at (-1,2) and transverse axis parallel to the -axis.also, a 2 = 1 and b 2 = 4, so c 2 = a 2 + b 2 = 5. Since the transverse axis is parallel to the -axis, the vertices and foci are located a and c units above and below the center, respectivel. The vertices are at (h,k ± a) = (-1,2 ± 1), or (-1,1) and (-1,3). The foci are at 1 (h,k ± c) = (-1,2 ± V5). The asmptotes are - 2 = 2'(x + 1) and 1-2 = -2'(x + 1). Figure 48(a) shows the graph drawn b hand. Figure 48(b) shows the graph obtained using a graphing utilit. Figure 48 (X + 1)2 ( - 2)2 - = 1 4 Transverse axis Y1 = 2 +J (x: 1) I/~->--':""""'~-+-""':"'~"-" I 5.1 V 1 = (-1,1) 5 X (a) (b) c..1ij!! r:r-- NOW WORK PROBLEM 45. Figure 49 Figure 50 d( P, ~) - d( P, F 2 ) = constant 6 Applications See Figure 49. Suppose that a gun is fired from an unknown source S. An observer at 0 1 hears the report (sound of gun shot) 1 second after another observer at O 2, Because sound travels at about 1100feet per second, it follows that the point S must be 1100feet closer to O 2 than to 0 1, S lies on one branch of a hperbola with foci at 0 1 and O 2, (Do ou see wh? The difference of the distances from S to 0 1 and from S to O 2 is the constant 1100.)If a third observer at 0 3 hears the same report 2 seconds after 0 1 hears it, then S will lie on a branch of a second hperbola with foci at 0 1 and 0 3, The intersection of the two hperbolas will pinpoint the location of S. Loran In the LOng RAnge Navigation sstem (LORAN), a master radio sending station and a secondar sending station emit signals that can be received b a ship at sea. See Figure 50. Because a ship monitoring the two signals will usuall be nearer to one of the two stations, there will be a difference in the distance that the two signals travel, which will register as a slight time difference between the signals. As long as the time difference remains constant, the difference of the two distances will also be constant. If the ship follows a path corresponding to the fixed time difference, it will follow the path of a hperbola whose foci are located at the positions of the two sending stations. So for each time difference a different hperbolic path results, each bringing the ship to a different shore location. Navigation charts show the various hperbolic paths corresponding to different time differences.

9 698 CHAPTER 9 Analtic Geometr EXAMPLE 9 LORAN Two LORAN stations are positioned 250 miles apart along a straight shore. (a) A ship records a time difference of second between the LORAN signals. Set up an appropriate rectangular coordinate sstem to determine where the ship would reach shore if it were to follow the hperbola corresponding to this time difference. (b) If the ship wants to enter a harbor located between the two stations 25 miles from the master station, what time difference should it be looking for? (c) If the ship is 80 miles offshore when the desired time difference is obtained, what is the approximate location of the ship? [Note: The speed of each radio signal is 186,000 miles per second.] Figure 51 ~(-io J 00 Solution seconds 15()- ::... (a) We set up a rectangular coordinate sstem so that the two stations lie on the x-axis and the origin is midwa between them. See Figure 51. The ship lies on a hperbola whose foci are the locations of the two stations. The reason for this is that the constant time difference of the signals from each station results in a constant difference in the distance of the ship from each station. Since the time difference is second and the speed of the signal is 186,000 miles per second, the difference of the distances from the ship to each station (foci) is Distance = Speed X Time = 186,000 X ::::> 100 miles I The difference of the distances from the ship to each station, 100, equals 2a, ~a = 50 and the vertex of the corresponding hperbola is at (50, 0). Since the focus is at (125,0), following this hperbola the ship would reach shore 75 miles from the master, station. (b) To reach shore 25 miles from the master station, the ship would follow a hperbola with vertex at (100,0). For this hperbola, a = 100, so the constant difference of the distances from the ship to each station is 2a = 200. The time difference that the ship should look for is T" Distance 200 ime = Speed = 186,000 ::::> second (c) To find the approximate location of the ship, we need to find the equation of the hperbola with vertex at (100,0) and a focus at (125,0). The form of the equation of this hperbola is X2 i ---=1 a 2 b 2 where a = 100. Since c = 125, we have b 2 = c 2 - a 2 = = 5625 The equation of the hperbola is X2 i = 1

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