944 Chapter 9 Conic Sections and Analtic Geometr 45. Use the polar equation for planetar orbits, to find the polar equation of the orbit for Mercur and Earth. Mercur: e = 0.056 and a = 36.0 * 10 6 miles Earth: e = 0.0167 and a = 9.96 * 10 6 miles Use a graphing utilit to graph both orbits in the same viewing rectangle. What do ou see about the orbits from their graphs that is not obvious from their equations? Critical Thinking Eercises Make Sense? In Eercises 46 49, determine whether each statement makes sense or does not make sense, and eplain our reasoning. 46. Eccentricit and polar coordinates enable me to see that ellipses, hperbolas, and parabolas are a unified group of interrelated curves. ep 47. I graphed a conic in the form that was 1 - e cos u smmetric with respect to the -ais. 48. Given the focus is at the pole, I can write the polar equation of a conic section if I know its eccentricit and the rectangular equation of the directri. 49. As long as I know how to graph in polar coordinates, a knowledge of conic sections is not necessar to graph the equations in Eercises 9 0. 50. Identif the conic and graph the equation: 11 - e a 1 - e cos u, 4 sec u sec u - 1. In Eercises 51 5, write a polar equation of the conic that is named and described. 51. Ellipse: a focus at the pole; verte: (4, 0); e = 1 5. Hperbola: a focus at the pole; directri: = -1; e = 3 53. Identif the conic and write its equation in rectangular 1 coordinates: - cos u. 54. Prove that the polar equation of a planet s elliptical orbit is 11 - e a 1 - e cos u, where e is the eccentricit and a is the length of the major ais. p q w r u Planet Sun: focus at pole a Preview Eercises Eercises 55 57 will help ou prepare for the material covered in the first section of the net chapter. 1-1 n 55. Evaluate for n = 1,, 3, and 4. 3 n - 1 56. Find the product of all positive integers from n down through 1 for n = 5. 57. Evaluate i + 1 for all consecutive integers from 1 to 6, inclusive. Then find the sum of the si evaluations. 0 Chapter Summar 9 Summar, Review, and Test DEFINITIONS AND CONCEPTS EXAMPLES 9.1 The Ellipse a. An ellipse is the set of all points in a plane the sum of whose distances from two fied points, the foci, is constant. b. Standard forms of the equations of an ellipse with center at the origin are [foci: 1-c, 0, 1c, 0] a + b = 1 E. 1, p. 876; E., p. 877; and [foci: 10, -c, 10, c], where c = a - b and a 7 b. See the bo on page 876 and b + a = 1 E. 3, p. 878 Figure 9.6. 1 - h 1 - k c. Standard forms of the equations of an ellipse centered at 1h, k are and a + b = 1 1 - h 1 - k See Table 9.1 on page 879. b + a = 1, a 7 b. E. 4, p. 880
DEFINITIONS AND CONCEPTS 9. The Hperbola a. A hperbola is the set of all points in a plane the difference of whose distances from two fied points, the foci, is constant. Summar, Review, and Test 945 b. Standard forms of the equations of a hperbola with center at the origin are [foci: 1-c, 0, 1c, 0] a - b = 1 E. 1, p. 888; E., p. 889 and [foci: 10, -c, 10, c], where c = a + b. See the bo on page 887 and Figure 9.16. a - b = 1 c. Asmptotes for are = ; b a Asmptotes for are = ; a - a - b = 1 a. b = 1 b. d. A procedure for graphing hperbolas is given in the lower bo on page 890. E. 3, p. 891; E. 4, p. 89 1 - h 1 - k e. Standard forms of the equations of a hperbola centered at 1h, k are and 1 - k 1 - h a - b = 1 See Table 9. on page 893. a - b = 1. 9.3 The Parabola a. A parabola is the set of all points in a plane that are equidistant from a fied line, the directri, and a fied point, the focus. EXAMPLES E. 5, p. 893; E. 6, p. 895 b. Standard forms of the equations of parabolas with verte at the origin are = 4p [focus: 1p, 0] and E. 1, p. 90; = 4p [focus: 10, p]. See the bo and Figure 9.31 on page 90. E. 3, p. 904 c. A parabola s latus rectum is a line segment that passes through its focus, is parallel to its directri, and has its endpoints on the parabola. The length of the latus rectum for = 4p and = 4p is ƒ 4p ƒ. A parabola can be graphed using the verte and endpoints of the latus rectum. E., p. 904 d. Standard forms of the equations of a parabola with verte at 1h, k are 1 - k = 4p1 - h and E. 4, p. 905; 1 - h = 4p1 - k. See Table 9.3 on page 905 and Figure 9.36. E. 5, p. 906 9.4 Rotation of Aes a. A nondegenerate conic section with equation of the form A + C + D + E + F = 0 in which A E. 1, p. 913 and C are not both zero is 1. a circle if A = C;. a parabola if AC = 0; 3. an ellipse if A Z C and AC 7 0; 4. a hperbola if AC 6 0. b. Rotation of Aes Formulas u is the angle from the positive -ais to the positive -ais. = cos u - sin u and = sin u + cos u c. Amount of Rotation Formula The general second-degree equation A + B + C + D + E + F = 0 can be rewritten in and without an -term b rotating the aes through angle u, where cot u = A - C and u is an acute angle. B E., p. 916 d. If u in cot u is one of the familiar angles such as 30, 45, or 60, write the equation of a rotated conic in E. 3, p. 918 standard form using the five-step procedure in the bo on page 918. e. If cot u is not the cotangent of one of the more familiar angles, use a sketch of cot u to find cos u. E. 4, p. 91 Then use sin u = A 1 - cos u to find values for sin u and cos u in the rotation formulas. f. A nondegenerate conic section of the form and cos u = A 1 + cos u A + B + C + D + E + F = 0 is 1. a parabola if B - 4AC = 0;. an ellipse or a circle if B - 4AC 6 0; 3. a hperbola if B - 4AC 7 0. E. 5, p. 93
946 Chapter 9 Conic Sections and Analtic Geometr DEFINITIONS AND CONCEPTS 9.5 Parametric Equations EXAMPLES a. The relationship between the parametric equations = f1t and = g1t and plane curves is described in the first bo on page 96. b. Point plotting can be used to graph a plane curve described b parametric equations. See the second bo on page 96. E. 1, p. 96 c. Plane curves can be sketched b eliminating the parameter t and graphing the resulting rectangular E., p. 97; equation. It is sometimes necessar to change the domain of the rectangular equation to be consistent with E. 3, p. 98 the domain for the parametric equation in. d. Infinitel man pairs of parametric equations can represent the same plane curve. One pair for = f1is E. 4, p. 930 = t and = f1t, in which t is in the domain of f. 9.6 Conic Sections in Polar Coordinates a. The focus-directri definitions of the conic sections are given in the bo on page 935. For all points on a conic, the ratio of the distance from a fied point (focus) and the distance from a fied line (directri) is constant and is called its eccentricit. If e = 1, the conic is a parabola. If e 6 1, the conic is an ellipse. If e 7 1, the conic is a hperbola. b. Standard forms of the polar equations of conics are ep and 1 ; e cos u ep 1 ; e sin u, in which 1r, u is a point on the conic s graph, e is the eccentricit, and p is the distance between the focus (located at the pole) and the directri. Details are shown in the bo on page 936. c. A procedure for graphing the polar equation of a conic is given in the bo on page 937. E. 1, p. 937; E., p. 939; E. 3, p. 940 Review Eercises 9.1 In Eercises 1 8, graph each ellipse and locate the foci. 1.. 5 + 36 + 5 = 1 16 = 1 3. 4 + = 16 4. 4 + 9 = 36 5. 1-1 1 + 1 + 1 1 - + = 1 6. + 16 9 9 16 7. 4 + 9 + 4-36 + 36 = 0 8. 9 + 4-18 + 8-3 = 0 = 1 13. A semielliptic archwa has a height of 15 feet at the center and a width of 50 feet, as shown in the figure. The 50-foot width consists of a two-lane road. Can a truck that is 1 feet high and 14 feet wide drive under the archwa without going into the other lane? 50 feet 15 feet 14 feet In Eercises 9 11, find the standard form of the equation of each ellipse satisfing the given conditions. 9. Foci: 1-4, 0, (4, 0); Vertices: 1-5, 0, (5, 0) 10. Foci: 10, -3, (0, 3); Vertices: 10, -6, (0, 6) 11. Major ais horizontal with length 1; length of minor ais = 4; center: 1-3, 5. 1. A semielliptical arch supports a bridge that spans a river 0 ards wide.the center of the arch is 6 ards above the river s center. Write an equation for the ellipse so that the -ais coincides with the water level and the -ais passes through the center of the arch. 0 d 6 d 14. An elliptical pool table has a ball placed at each focus. If one ball is hit toward the side of the table, eplain what will occur. 9. In Eercises 15, graph each hperbola. Locate the foci and find the equations of the asmptotes. 1 feet 15. 16 - = 1 16. 16 - = 1 17. 9-16 = 144 18. 4 - = 16 19. 1-1 + 3 1 + 1-3 - = 1 0. - 5 16 5 16 = 1 1. - 4-4 + 8-4 = 0. - - - - 1 = 0
Summar, Review, and Test 947 In Eercises 3 4, find the standard form of the equation of each hperbola satisfing the given conditions. 3. Foci: 10, -4, (0, 4); Vertices: 10, -, (0, ) 4. Foci: 1-8, 0, (8, 0); Vertices: 1-3, 0, (3, 0) 5. Eplain wh it is not possible for a hperbola to have foci at 10, - and (0, ) and vertices at 10, -3 and (0, 3). 6. Radio tower M is located 00 miles due west of radio tower M 1. The situation is illustrated in the figure shown, where a coordinate sstem has been superimposed. Simultaneous radio signals are sent from each tower to a ship, with the signal from M received 500 microseconds before the signal from M 1. Assuming that radio signals travel at 0.186 mile per microsecond, determine the equation of the hperbola on which the ship is located. 9.3 M 00 miles M 1 ( 100, 0) (100, 0) In Eercises 7 33, find the verte, focus, and directri of each parabola with the given equation. Then graph the parabola. 7. = 8 8. + 16 = 0 9. 1 - = -16 30. 1-4 = 41 + 1 31. + 4 = 4 3. - 4-10 + 1 = 0 33. - 4 - = 0 In Eercises 34 35, find the standard form of the equation of each parabola satisfing the given conditions. 34. Focus: (1, 0); Directri: = -1 35. Focus: 10, -11; Directri: = 11 36. An engineer is designing headlight units for automobiles. The unit has a parabolic surface with a diameter of 1 inches and a depth of 3 inches. The situation is illustrated in the figure, where a coordinate sstem has been superimposed. What is the equation of the parabola in this sstem? Where should the light source be placed? Describe this placement relative to the verte. 6 6 Verte (0, 0) P 1 inches (6, 3) 3 inches 37. The George Washington Bridge spans the Hudson River from New York to New Jerse. Its two towers are 3500 feet apart and rise 316 feet above the road. As shown in the figure, the cable between the towers has the shape of a parabola and the cable just touches the sides of the road midwa between the towers. What is the height of the cable 1000 feet from a tower? 38. The giant satellite dish in the figure shown is in the shape of a parabolic surface. Signals strike the surface and are reflected to the focus, where the receiver is located.the diameter of the dish is 300 feet and its depth is 44 feet. How far, to the nearest foot, from the base of the dish should the receiver be placed? 9.4 Parabolic Cable 1750 3500 feet (1750, 316) 1750 300 feet Receiver (150, 44) 44 feet In Eercises 39 46, identif the conic represented b each equation without completing the square or using a rotation of aes. 39. + 4 + - 15 = 0 40. + 16-160 + 384 = 0 41. 16 + 64 + 9-54 + 1 = 0 4. 4-9 - 8 + 1-144 = 0 43. 5 + 3 + 3-18 = 0 44. 5-8 + 7-95 - 9 = 0 45. + 6 + 9 - = 0 46. - + 3 + + 4-1 = 0 In Eercises 47 51, a. Rewrite the equation in a rotated -sstem without an -term. b. Epress the equation involving and in the standard form of a conic section. c. Use the rotated sstem to graph the equation. 47. - 4 = 0 48. + + - 1 = 0 49. 4 + 10 + 4-9 = 0 50. 6-6 + 14-45 = 0 51. + 3 + 3-13 + 1 = 0
948 Chapter 9 Conic Sections and Analtic Geometr 9.5 In Eercises 5 57, eliminate the parameter and graph the plane curve represented b the parametric equations. Use arrows to show the orientation of each plane curve. 5. 53. 54. 55. 56. = t - 1, = 1 - t; - q 6 t 6 q = t, = t - 1; -1 t 3 = 4t, = t + 1; - q 6 t 6 q = 4 sin t, = 3 cos t; 0 t 6 p = 3 + cos t, = 1 + sin t; 0 t 6 p 57. = 3 sec t, = 3 tan t; 0 t p 4 58. Find two different sets of parametric equations for = + 6. 59. The path of a projectile that is launched h feet above the ground with an initial velocit of v 0 feet per second and at an angle u with the horizontal is given b the parametric equations = 1v 0 cos ut and = h + 1v 0 sin ut - 16t, where t is the time, in seconds, after the projectile was launched. A football plaer throws a football with an initial velocit of 100 feet per second at an angle of 40 to the horizontal. The ball leaves the plaer s hand at a height of 6 feet. a. Find the parametric equations that describe the position of the ball as a function of time. 9.6 b. Describe the ball s position after 1,, and 3 seconds. Round to the nearest tenth of a foot. c. How long, to the nearest tenth of a second, is the ball in flight? What is the total horizontal distance that it travels before it lands? d. Graph the parametric equations in part (a) using a graphing utilit. Use the graph to determine when the ball is at its maimum height. What is its maimum height? Round answers to the nearest tenth. In Eercises 60 65, a. If necessar, write the equation in one of the standard forms for a conic in polar coordinates. b. Determine values for e and p. Use the value of e to identif the conic section. c. Graph the given polar equation. 60. 4 1 - sin u 61. 6. 6 + sin u 63. 64. 6 3 + 6 sin u 65. 6 1 + cos u 3 - cos u 8 4 + 16 cos u Chapter 9 Test In Eercises 1 5, graph the conic section with the given equation. For ellipses, find the foci. For hperbolas, find the foci and give the equations of the asmptotes. For parabolas, find the verte, focus, and directri. 4 feet 1. 9-4 = 36. 3. 4. 5. = -8 1 + 5 1-5 + = 1 9 4 - + 8 + + 7 = 0 1 + 5 = 81-1 In Eercises 6 8, find the standard form of the equation of the conic section satisfing the given conditions. 10. An engineer is designing headlight units for cars. The unit shown in the figure below has a parabolic surface with a diameter of 6 inches and a depth of 3 inches. 80 feet 6. Ellipse; Foci: 1-7, 0, (7, 0); Vertices: 1-10, 0, (10, 0) 7. Hperbola; Foci: 10, -10, (0, 10); Vertices: 10, -7, (0, 7) 8. Parabola; Focus: (50, 0); Directri: = -50 9. A sound whispered at one focus of a whispering galler can be heard at the other focus. The figure at the top of the net column shows a whispering galler whose cross section is a semielliptical arch with a height of 4 feet and a width of 80 feet. How far from the room s center should two people stand so that the can whisper back and forth and be heard? 3 3 Verte (0, 0) 6 inches 3 inches
Summar, Review, and Test 949 a. Using the coordinate sstem that has been positioned on the unit, find the parabola s equation. b. If the light source is located at the focus, describe its placement relative to the verte. In Eercises 11 1, identif each equation without completing the square or using a rotation of aes. 11. + 9 + 10-18 + 5 = 0 1. + + + 3 - - 3 = 0 13. For the equation In Eercises 14 15, eliminate the parameter and graph the plane curve represented b the parametric equations. Use arrows to show the orientation of each plane curve. 14. = t, = t - 1; - q 6 t 6 q 15. = 1 + 3 sin t, = cos t; 0 t 6 p In Eercises 16 17, identif the conic section and graph the polar equation. 4 16. 17. 1 - cos u + sin u 7-63 + 13-16 = 0, determine what angle of rotation would eliminate the -term in a rotated -sstem. Cumulative Review Eercises (Chapters P 9) ƒ ƒ Solve each equation or inequalit in Eercises 1 7. 1.. 3. 1-3 + 5 = 81-1 -31-4 7 16-1 - 5 = + 7 4. 1 - = 0 5. - 1 Ú 7 6. 3 3 + 4-7 + = 0 7. log 1 + 1 + log 1-1 = 3 Solve each sstem in Eercises 8 10. 3 + 4 = 8. b + 5 = -1 10. (Use matrices.) 9. b - = -8 - = 6 In Eercises 11 13, graph each equation, function, or sstem in a rectangular coordinate sstem. 11. f1 = 1-1. 9 + 1-4 4 = 1 13. 5 + 10 c Ú 1 4 + - + z = 17 c -4 + + 5z = - + 3 + z = 8 14. a. List all possible rational roots of 3 3-5 + 17 + 3 = 0. b. The graph of f1 = 3 3-5 + 17 + 3 is shown in a 3-1, 3, 14 b 3-, 6, 14 viewing rectangle. Use the graph of f and snthetic division to solve the equation in part (a). 15. The figure shows the graph of = f1 and its two vertical asmptotes. 5 4 3 1 5 4 3 1 1 3 4 5 = f() 1 3 4 5 a. Find the domain and the range of f. b. What is the relative minimum and where does it occur? c. Find the interval on which f is increasing. d. Find f1-1 - f10. e. Find 1f f11. f. Use arrow notation to complete this statement: f1 : q as or as. g. Graph g1 = f1 - + 1. h. Graph h1 = -f1. 16. If f1 = - 4 and g1 = +, find 1g f1. 17. Epand using logarithmic properties. Where possible, evaluate logarithmic epressions. log 5 3 1 15 18. Write the slope-intercept form of the equation of the line passing through 11, -4 and 1-5, 8.
950 Chapter 9 Conic Sections and Analtic Geometr 19. Rent-a-Truck charges a dail rental rate for a truck of $39 plus $0.16 a mile. A competing agenc, Ace Truck Rentals, charges $5 a da plus $0.4 a mile for the same truck. How man miles must be driven in a da to make the dail cost of both agencies the same? What will be the cost? 0. The local cable television compan offers two deals. Basic cable service with one movie channel costs $35 per month. Basic service with two movie channels cost $45 per month. Find the charge for the basic cable service and the charge for each movie channel. csc u - sin u 1. Verif the identit: = cot u. sin u. Graph one complete ccle of = cos1 + p. 3. If v = 3i - 6j and w = i + j, find 1v # ww. 4. Solve for u: sin u = sin u, 0 u 6 p. 5. In oblique triangle ABC, A = 64, B = 7, and a = 13.6. Solve the triangle. Round lengths to the nearest tenth.