QUADRATIC RELATIONS AND CONIC SECTIONS. What shape does a telescope mirror have?

Transcription

1 QUADRATIC RELATIONS AND CONIC SECTIONS What shape does a telescope mirror have? 586

2 APPLICATION: Telescopes Using powerful telescopes like the Hubble telescope, astronomers have discovered planets that eist outside our solar system. By eamining the wobble in a star s motion, astronomers can detect the presence of one or more planets orbiting that star. The challenge is to see the planets, but that would require a telescope with a mirror 100 meters in diameter, 10 times larger than any eisting telescope. Think & Discuss C H A P T E R 10 The diagram shows a cross section of a mirror from a telescope at the Big Bear Solar Observatory in California. An equation for the surface of the mirror, based on the coordinate system shown, is y = where and y are measured in centimeters. 1. What shape is the cross section of the mirror?. The mirror has a diameter of 65 cm. What is the depth of the mirror? How did you get your answer? Learn More About It In Eercise 67 on p. 630 you will use your knowledge of conic sections to determine what shape a telescope s mirrors have. INTERNET APPLICATION LINK Visit for more information about telescopes. 587

3 CHAPTER 10 Study Guide PREVIEW What s the chapter about? Chapter 10 is about conic sections. The four conic sections are parabolas, circles, ellipses, and hyperbolas. In Chapter 10 you ll learn how to use the distance and midpoint formulas. how to graph and write equations of conics, and how to classify conics. how to solve systems of quadratic equations. KEY VOCABULARY Review parabola, p. 49 hyperbola, p. 540 New distance formula, p. 589 midpoint formula, p. 590 circle, p. 601 ellipse, p. 609 hyperbola, p. 615 conic sections, p. 63 general second-degree equation, p. 66 discriminant, p. 66 PREPARE Are you ready for the chapter? SKILL REVIEW Do these eercises to review key skills that you ll apply in this chapter. See the given reference page if there is something you don t understand. STUDENT HELP Study Tip Student Help boes throughout the chapter give you study tips and tell you where to look for etra help in this book and on the Internet. Write an equation of the line that passes through the given point and has the given slope. (Review Eample, p. 9) 1. (0, 4), m =. (, º), m = (º4, 1), m = º 3 4 Solve the system using any algebraic method. (Review Eamples 1 3, pp ) 4. + y = y = º y = 7 3 º y = y = 5 º y = Graph the function. Label the verte and ais of symmetry. (Review Eamples 1 3, pp. 50 and 51) 7. y = y = º3 9. y = ( º 3) º 1 Solve the equation by completing the square. (Review Eamples and 3, p. 83) = = º5 1. º = º STUDY STRATEGY Here s a study strategy! Dictionary of Graphs Make a dictionary of graphs to use as a reference tool. Draw and label an eample of each conic. Note the important characteristics of the conic, and write the conic s equation. Epand your dictionary to include all the types of graphs you have learned and continue to learn in this course. 588 Chapter 10

4 E X P L O R I N G D ATA A N D S TAT I S T I C S 10.1 What you should learn GOAL 1 Find the distance between two points and find the midpoint of the line segment joining two points. GOAL Use the distance and midpoint formulas in real-life situations, such as finding the diameter of a broken dish in Eample 5. Why you should learn it To solve real-life problems, such as finding the distance a medical helicopter must travel to a hospital in Es REAL LIFE The Distance and Midpoint Formulas GOAL 1 USING THE DISTANCE AND MIDPOINT FORMULAS To find the distance d between A( 1, y 1 ) and B(, y ), you can apply the Pythagorean theorem to right triangle ABC. (AB) = (AC) + (BC) d = ( º 1 ) + (y º y 1 ) d = ( º 1 ) + ( y º y 1 ) The third equation is called the THE DISTANCE FORMULA E X A M P L E 1 distance formula. The distance d between the points ( 1, y 1 ) and (, y ) is as follows: d = ( º 1 ) + ( y º y 1 ) Finding the Distance Between Two Points Find the distance between (º, 5) and (3, º1). d B(, y ) A( 1, y 1 ) C(, y 1 ) Let ( 1, y 1 ) = (º, 5) and (, y ) = (3, º1). d = ( º 1 ) + ( y º y 1 ) = (3 º ( º )) + ( º 1 º 5 ) Substitute. = Simplify. Use distance formula. = Use a calculator. E X A M P L E Classifying a Triangle Using the Distance Formula Classify ABC as scalene, isosceles, or equilateral. A(4, 6) AB = (6 º 4 ) + ( 1 º 6 ) = 9 BC = (1 º 6 ) + ( 3 º 1 ) = 9 AC = (1 º 4 ) + ( 3 º 6 ) = 3 Because AB = BC, ABC is isosceles. C (1, 3) B(6, 1) 10.1 The Distance and Midpoint Formulas 589

5 Another formula involving two points in a coordinate plane is the midpoint formula. Recall that the midpoint of a segment is the point on the segment that is equidistant from the two endpoints. THE MIDPOINT FORMULA The midpoint of the line segment joining A( 1, y 1 ) and B(, y ) is as follows: M 1 +, y 1 + y Each coordinate of M is the mean of the corresponding coordinates of A and B. A( 1, y 1 ) midpoint M B(, y ) E X A M P L E 3 Finding the Midpoint of a Segment Find the midpoint of the line segment joining (º7, 1) and (º, 5). Let ( 1, y 1 ) = (º7, 1) and (, y ) = (º, 5). 1 +, y 1 + y º7 + (º) =, = º 9, 3 9, 3 ( 7, 1) (, 5) E X A M P L E 4 Finding a Perpendicular Bisector STUDENT HELP Look Back For help with perpendicular lines, see p. 9. Write an equation for the perpendicular bisector of the line segment joining A(º1, 4) and B(5, ). First find the midpoint of the line segment: 1 +, y 1 + y = º1 + 5, 4 + = (, 3) Æ Then find the slope of AB : m = y º y1 º 4 = = º = º 1 º 5 º (º 1) The slope of the perpendicular bisector is the negative reciprocal of º 1, or mfi = 3. 3 Since you know the slope of the perpendicular bisector and a point that the bisector passes through, you can use the point-slope form to write its equation. y º 3 = 3( º ) y = 3 º 3 A( 1, 4) Æ An equation for the perpendicular bisector of AB is y = 3 º 3. (, 3) B(5, ) y Chapter 10 Quadratic Relations and Conic Sections

6 !!!! " " FOCUS ON APPLICATIONS GOAL DISTANCE AND MIDPOINT FORMULAS IN REAL LIFE Recall from geometry that the perpendicular bisector of a chord of a circle passes through the center of the circle. Using this theorem, you can find the center of a circle given three points on the circle. E X A M P L E 5 Using the Distance and Midpoint Formulas in Real Life ARCHEOLOGISTS use grids to systematically eplore a site. By labeling the grid squares, they can record where each artifact is found. REAL LIFE ARCHEOLOGY While on an archeological dig, you discover a piece of a broken dish. To estimate the original diameter of the dish, you lay the piece on a coordinate plane and mark three points on the circular edge, as shown. Use these points to find the diameter of the dish. (Each unit in the coordinate plane represents 1 inch.) Use the method illustrated in Eample 4 to find the perpendicular bisectors of AO Æ Æ and OB. y = + 5 y = º Æ Perpendicular bisector of AO Æ Perpendicular bisector of OB A ( 4, ) O(0, 0) B (6, 4) Both bisectors pass through the circle s center. Therefore, the center of the circle is the solution of the system formed by these two equations. y = + 5 Write first equation. º = + 5 Substitute for y. C º = Multiply each side by. B º7 = º3 Simplify. A = 3 Divide each side by º7. 7 O y = Substitute the -value into the first equation. y = 4 71 Simplify. STUDENT HELP Look Back For help with solving systems, see p The center of the circle is C 3 7, The radius of the circle is the distance between C and any of the three given points. CO = 0 º º = The dish had a diameter of about (5.87) = inches The Distance and Midpoint Formulas 591

7 " # $ % & ' ( ) * GUIDED PRACTICE Vocabulary Check Concept Check Skill Check 1. State the distance and midpoint formulas.. Look back at Eample 1. Find the distance between (º, 5) and (3, º1), but this time letting ( 1, y 1 ) = (3, º1) and (, y ) = (º, 5). How are the calculations different? Do you get the same answer? 3. a. Write a formula for the distance between a point (, y) and the origin. b. Write a formula for the midpoint of the segment joining a point (, y) and the origin. Find the distance between the two points. 4. (, º1), (, 3) 5. (º5, º), (0, º) 6. (0, 6), (4, 9) 7. (10, º), (7, 4) 8. (º3, 8), (5, 6) 9. (6, º1), (º9, 8) Find the midpoint of the line segment joining the two points. 10. (0, 0), (º8, 14) 11. (0, 3), (4, 9) 1. (1, º), (1, 6) 13. (1, 3), (3, 11) 14. (º5, 4), (, º4) 15. (º1, 5), (º8, º6) 16. HIKING You are going on a two-day hike. The map at the right shows the trails you plan to follow. (Each unit represents 1 mile.) a. You hike from the lodge to point A and decide Æ that you will hike to the midpoint of AB before you camp for the night. At what point in the plane will you be camping? b. How far will you hike each day? A( 3, ) B(, 6) lodge (0, 0) PRACTICE AND APPLICATIONS STUDENT HELP Etra Practice to help you master skills is on p STUDENT HELP HOMEWORK HELP Eample 1: Es , Eample : Es Eample 3: Es Eample 4: Es Eample 5: Es USING THE FORMULAS Find the distance between the two points. Then find the midpoint of the line segment joining the two points. 17. (0, 0), (3, 4) 18. (0, 0), (4, 1) 19. (0, 4), (8, º3) 0. (º, 8), (6, 0) 1. (º3, º1), (7, 4). (9, º), (3, 6) 3. (º5, º8), (1, 6) 4. (º, 10), (10, º) 5. (8, 3), (, º1) 6. (º10, º15), (1, 18) 7. (º3.5, 1.), (6, º3.8) 8. (6.3, º9), (1.3, º8.5) 9. (º7, ), º 1 1, , º 1 4 1, º 7, º º 3 4,, 5, º 7 4 GEOMETRY CONNECTION The vertices of a triangle are given. Classify the triangle as scalene, isosceles, or equilateral. 3. (, 0), (0, 8), (º, 0) 33. (4, 1), (1, º), (6, º4) 34. (1, 9), (º4, ), (4, ) 35. (, 5), (8, ), (4, º1) 36. (5, º1), (º4, 0), (3, 5) 37. (4, 4), (8, 1), (6, º5) 38. (0, º3), (3, 5), (º5, ) 39. (1, 1), (º4, 0), (º, 5) 40. (, 4), (3, º), (º1, 1) 59 Chapter 10 Quadratic Relations and Conic Sections

8 !! + ", # $ % & ' ( ) * FINDING EQUATIONS Write an equation for the perpendicular bisector of the line segment joining the two points. 41. (, ), (6, 14) 4. (0, 0), (º8, º10) 43. (0, º6), (º4, 9) 44. (3, º7), (º3, 1) 45. (º3, º7.), (º4., 1.8) 46. 3, º6, (º3, 1) FINDING A COORDINATE Use the given distance d between the two points to solve for. 47. (0, 1), (, 4); d = (1, 3), (º6, ); d = (, º10), (º8, 4); d = (0.5, ), (7, ); d = 8.5 ACCIDENT RECONSTRUC- TIONIST An accident reconstructionist uses physical evidence, such as skid marks, to determine how accidents occurred. REAL CAREER LINK INTERNET STUDENT HELP HOMEWORK HELP Visit our Web site for help with problem solving in Es. 51 and 5. INTERNET LIFE FOCUS ON CAREERS URBAN PLANNING In Eercises 51 and 5, use the following information. You are designing a city park like the one shown at the right. You want the park to have two fountains so that each fountain is equidistant from four of the si park entrances. The labeled points shown in the coordinate plane represent the park entrances. 51. Where should the fountains be placed? 5. How far apart should the fountains be placed? HELICOPTER RESCUE In Eercises 53 56, use the following information to find the distance a medical helicopter would have to travel to St. John s Hospital from each highway intersection. The Highway Department of Sangamon County in Illinois uses a map with a coordinate plane whose origin represents downtown Springfield. Each unit represents one mile and the letters N, S, E, and W are used to indicate the direction. For eample, 3E 5S corresponds to (3, º5), a point 3 miles east and 5 miles south of downtown Springfield. St. John s Hospital is located at 1E 0, or (1, 0). 53. Rt. 1 Rt. 3 intersection at 19E 6N 54. Rt. 37 Rt. 40 intersection at 6E 9S 55. Rt. 18 Rt. 40 intersection at 6W 9S 56. Rt. 10 Rt. 47 intersection at 14W 1N 57. ACCIDENT RECONSTRUCTION When a car makes a fast, sharp turn, an accident reconstructionist can use the car s skid mark to determine its speed. The equation v = a r gives the car s speed v (in meters per second) as a function of the radius r of the circle (in meters) along which the car was traveling. The constant a (measured in meters per second squared) varies depending on road conditions. Find the radius of the skid mark shown below. Then use the given equation and 6.86 for a to find how fast the car was going. Source: Mathematical Modeling ( 10, 1) (0, 0) 10 14W 1N W 9S 40 1 St. John's Hospital 1E E 9S (16, ) 19E 6N The Distance and Midpoint Formulas 593

9 # $ % & ' ( ) * 58. STATISTICS CONNECTION A physician uses many tests to evaluate a patient s condition. Some of these tests yield numerical results. In these cases, the physician can treat two test results as an ordered pair and use the distance formula to determine how close to average the patient is. In the table below the serum creatinine (C) and systolic blood pressure (P) for several patients are given. Tell how far from normal each patient is, where normal is represented by the ordered pair (C, P) = (1, 17). Test Preparation Challenge QUANTITATIVE COMPARISON In Eercises 59º6, choose the statement that is true about the given quantities C P A The quantity in column A is greater. B The quantity in column B is greater. C The two quantities are equal. D The relationship cannot be determined from the given information. Column A Column B Distance between (0, 7) and (1, º1) Distance between (9, ) and (3, 8) Distance between (º5, º) and (5, ) Distance between (º5, 5) and (, º) Distance between (º3, 0) and (, º4) Distance between (7, 6) and (1, 5) Distance between (, º5) and (1, 6) Distance between (0, 8) and (6, 0) 63. FINDING A FORMULA Find formulas for the distance between a point (, y) and each of the following: (a) a horizontal line y = k and (b) a vertical line = h. MIXED REVIEW GRAPHING FUNCTIONS Graph the quadratic function. (Review 5.1 for 10.) 64. y = y = y = º3 67. y = º 68. y = y = º y = º y = 5 6 SOLVING EQUATIONS Solve the equation. Check for etraneous solutions. (Review 7.6) 7. / = = = º = = º 3/ = º8 OPERATIONS WITH RATIONAL EXPRESSIONS Perform the indicated operation and simplify. (Review 9.5) º º º 1 3 4( º 5) º º º º 3 + º Chapter 10 Quadratic Relations and Conic Sections

10 , 6 -. / Parabolas What you should learn GOAL 1 Graph and write equations of parabolas. GOAL Use parabolas to solve real-life problems, such as finding the depth of a solar energy collector in Eample 3. Why you should learn it To model real-life parabolas, such as the reflector of a car s headlight in E. 79. REAL LIFE GOAL 1 GRAPHING AND WRITING EQUATIONS OF PARABOLAS You already know that the graph of y = a is a parabola whose verte (0, 0) lies on its ais of symmetry = 0. Every parabola has the property that any point on it is equidistant from a point called the focus and a line called the directri. The focus lies on the ais of symmetry. The verte lies halfway between the focus and the directri. The directri is perpendicular to the ais of symmetry. In Chapter 5 you saw parabolas that have a vertical ais of symmetry and open up or down. In this lesson you will also work with parabolas that have a horizontal ais of symmetry and open left or right. In the four cases shown below, the focus and the directri each lie p units from the verte. focus: (0, p) verte: (0, 0) directri: y p verte: (0, 0) directri: y p focus: (0, p) 4py, p > 0 4py, p < 0 directri: p directri: p focus: (p, 0) focus: (p, 0) verte: (0, 0) verte: (0, 0) y 4p, p > 0 y 4p, p < 0 Characteristics of the parabolas shown above are given on the net page. 10. Parabolas 595

11 B? A : ; < = > STANDARD EQUATION OF A PARABOLA ( VERTEX AT ORIGIN) The standard form of the equation of a parabola with verte at (0, 0) is as follows. EQUATION FOCUS DIRECTRIX AXIS OF SYMMETRY = 4py (0, p) y = ºp Vertical ( = 0) y = 4p (p, 0) = ºp Horizontal (y = 0) E X A M P L E 1 Graphing an Equation of a Parabola STUDENT HELP Look Back For help with drawing parabolas, see p. 49. Identify the focus and directri of the parabola given by = º 1 6 y. Draw the parabola. Because the variable y is squared, the ais of symmetry is horizontal. To find the focus and directri, rewrite the equation as follows. = º 1 6 y Write original equation. º6 = y Multiply each side by º6. Since 4p = º6, you know p = º 3. The focus is (p, 0) = º 3, 0 and the directri is = ºp = 3. To draw the parabola, make a table of values and plot points. Because p < 0, the parabola opens to the left. Therefore, only negative -values should be chosen. 3, 0 3? A º1 º º3 º4 º5 y ±.45 ±3.46 ±4.4 ±4.90 ±5.48 E X A M P L E Writing an Equation of a Parabola STUDENT HELP Write an equation of the parabola shown at the right. HOMEWORK HELP Visit our Web site for etra eamples. INTERNET The graph shows that the verte is (0, 0) and the directri is y = ºp = º. Substitute for p in the standard equation for a parabola with a vertical ais of symmetry. verte directri = 4py Standard form, vertical ais of symmetry = 4()y Substitute for p. = 8y Simplify. CHECK You can check this result by solving the equation for y to get y = 1 8 and graphing the equation using a graphing calculator. 596 Chapter 10 Quadratic Relations and Conic Sections

12 7 8 9 : ; < = > GOAL USING PARABOLAS IN REAL LIFE Parabolic reflectors have cross sections that are parabolas. A special property of any parabolic reflector is that all incoming rays parallel to the ais of symmetry that hit the reflector are directed to the focus (Figure 1). Similarly, rays emitted from the focus that hit the reflector are directed in rays parallel to the ais of symmetry (Figure ). These properties are the reason satellite dishes and flashlights are parabolic. Figure 1 Figure E X A M P L E 3 Modeling a Parabolic Reflector SOLAR ENERGY Sunfire is a glass parabola used to collect solar energy. The sun s rays are reflected from the mirrors toward two boilers located at the focus of the parabola. When heated, the boilers produce steam that powers an alternator to produce electricity. a. Write an equation for Sunfire s cross section. b. How deep is the dish? boiler Sunfire 10 ft depth FOCUS ON PEOPLE 37 ft a. The boilers are 10 feet above the verte of the dish. Because the boilers are at the focus and the focus is p units from the verte, you can conclude that p = 10. Assuming the verte is at the origin, an equation for the parabolic cross section is as follows: = 4py Standard form, vertical ais of symmetry HOWARD FRANK BROYLES is an engineer and inventor. More than 50 high school students volunteered their time to help him build Sunfire. The project took over ten years. REAL APPLICATION LINK INTERNET LIFE = 4(10)y Substitute 10 for p. = 40y Simplify. b. The dish etends 3 7 = 18.5 feet on either side of the origin. To find the depth of the dish, substitute 18.5 for in the equation from part (a). = 40y Equation for the cross section (18.5) = 40y Substitute 18.5 for. 8.6 y Solve for y. The dish is about 8.6 feet deep. 10. Parabolas 597

13 M M K K M M L L M N K K M N L L M K K M L L C D E F G H I J GUIDED PRACTICE Vocabulary Check Concept Check Skill Check 1. Complete this statement: A parabola is the set of points equidistant from a point called the? and a line called the?.. How does the graph of = ay differ from the graph of y = a? 3. Knowing the value of a in y = a, how can you find the focus and directri? Graph the equation. Identify the focus and directri of the parabola. 4. = 4y 5. y = º5 6. º1 = y 7. 8y = 8. º6 = y 9. = y Write the standard form of the equation of the parabola with the given focus or directri and verte at (0, 0). 10. focus: (0, 3) 11. focus: (5, 0) 1. focus: (º6, 0) 13. directri: = directri: = º1 15. directri: y = 8 PRACTICE AND APPLICATIONS STUDENT HELP Etra Practice to help you master skills is on p MATCHING Match the equation with its graph. 16. y = = º4y 18. = 4y 19. y = º4 0. y = = 1 4 y A. B. C. M D. E. F. M DIRECTION Tell whether the parabola opens up, down, left, or right.. y = º3 3. º9 = y 4. y = º6 5. = 7y STUDENT HELP HOMEWORK HELP Eample 1: Es Eample : Es Eample 3: Es = 16y 7. º3y = 8 8. º5 = ºy 9. = 4 3 y FOCUS AND DIRECTRIX Identify the focus and directri of the parabola = ºy 31. y = 3. = 8y 33. y = º y = º = º36y 36. º4 + 9y = º8y + = Chapter 10 Quadratic Relations and Conic Sections

14 O P Q R S T U V GRAPHING Graph the equation. Identify the focus and directri of the parabola. 38. y = = º6y 40. y = º 41. y = 4 4. = 8y 43. y = º = º0y 45. = 18y 46. = º4y 47. = 16y 48. y = y = º º 40y = y 0 = = 4y 53. º 1 8 y = 0 WRITING EQUATIONS Write the standard form of the equation of the parabola with the given focus and verte at (0, 0). 54. (4, 0) 55. (º, 0) 56. (º3, 0) 57. (0, 1) 58. (0, 4) 59. (0, º3) 60. (0, º4) 61. (º5, 0) 6. º 1 4, , º , , 0 WRITING EQUATIONS Write the standard form of the equation of the parabola with the given directri and verte at (0, 0). 66. y = 67. y = º3 68. = º4 69. = = º5 71. y = º1 7. = 73. y = = º = y = y = º COMMUNICATIONS The cross section of a television antenna dish is a parabola. For the dish at the right, the receiver is located at the focus, 4 feet above the verte. Find an equation for the cross section of the dish. (Assume the verte is at the origin.) If the dish is 8 feet wide, how deep is it? 8 ft 4 ft MATH IN HISTORY Man has been aware of the reflective properties of parabolas for two thousand years. The above illustration of Archimedes burning mirror was taken from a book printed in the 17th century. REAL FOCUS ON APPLICATIONS LIFE 79. AUTOMOTIVE ENGINEERING The filament of a lightbulb is a thin wire that glows when electricity passes through it. The filament of a car headlight is at the focus of a parabolic reflector, which sends light out in a straight beam. Given that the filament is 1.5 inches from the verte, find an equation for the cross section of the reflector. If the reflector is 7 inches wide, how deep is it? 80. HISTORY CONNECTION In the drawing shown at the left, the rays of the sun are lighting a candle. If the candle flame is 1 inches from the back of the parabolic reflector and the reflector is 6 inches deep, then what is the diameter of the reflector? 81. CAMPING You can make a solar hot dog cooker using foil-lined cardboard shaped as a parabolic trough. The drawing at the right shows how to suspend a hot dog with a wire through the focus of each end piece. If the trough is 1 inches wide and 4 inches deep, how far from the bottom should the wire be placed? Source: Boys Life 7 in. 1.5 in. 1 in. 4 in. 10. Parabolas 599

15 W X O P Q R S T U V Test Preparation Challenge EXTRA CHALLENGE 8. Writing For an equation of the form y = a, discuss what effect increasing a has on the focus and directri. 83. MULTI-STEP PROBLEM A flashlight has a parabolic reflector. An equation for the cross section of the reflector is y = 3. The depth of the reflector is 3 7 inches. a. Writing Eplain why the value of p must be less than the depth of the reflector of a flashlight. filament bulb b. How wide is the beam of light projected by the flashlight? c. Write an equation for the cross section of a reflector having the same depth but a wider beam than the flashlight shown. How wide is the beam of the new reflector? d. Write an equation for the cross section of a reflector having the same depth but a narrower beam than the flashlight shown. How wide is the beam of the new reflector? 84. GEOMETRY CONNECTION The latus rectum of a parabola is the line segment that is parallel to the directri, passes through the focus, and has endpoints that lie on the parabola. Find the length of the latus rectum of a parabola given by = 4py. latus rectum reflector 4py y p MIXED REVIEW LOGARITHMIC AND EXPONENTIAL EQUATIONS Solve the equation. Check for etraneous solutions. (Review 8.6) = = = = log 7 (3 º 5) = log log 3 (4 º 3) = 3 OPERATIONS WITH RATIONAL EXPRESSIONS Perform the indicated operation and simplify. (Review 9.4 and 9.5) y y y º 9 ( + ) º º 6 FINDING A DISTANCE Find the distance between the two points. (Review 10.1 for 10.3) y 3 º º 1 6 º 96. º 3 + º º º 1 º 97. (3, 4), (6, 7) 98. (º3, 7), (º7, 3) 99. (18, º4), (º, 9) 100. (3.7, 5.1), (, 5) 101. (º9, º31), (8, 7) 10. (8.8, 3.3), (1., 6) 103. CONSUMER ECONOMICS The amount A (in dollars) you pay for grapes varies directly with the amount P (in pounds) that you buy. Suppose you buy 1 1 pounds for $.5. Write a linear model that gives A as a function of P. (Review.4) y Chapter 10 Quadratic Relations and Conic Sections

16 e c c a b d Y Z [ \ ] ^ _ ` 10.3 Circles What you should learn GOAL 1 Graph and write equations of circles. GOAL Use circles to solve real-life problems, such as determining whether you are affected by an earthquake in E. 81. Why you should learn it To model real-life situations with circular models, such as the region lit by a lighthouse beam in Eample 4. REAL LIFE GOAL 1 GRAPHING AND WRITING EQUATIONS OF CIRCLES A circle is the set of all points (, y) that are equidistant from a fied point, called the center of the circle. The distance r between the center and any point (, y) on the circle is the radius. The distance formula can be used to obtain an equation of the circle whose center is the origin and whose radius is r. Because the distance between any point (, y) on the circle and the center (0, 0) is r, you can write the following. ( º 0 ) + ( y º 0 ) = r Distance formula ( º 0) + (y º 0) = r Square both sides. + y = r Simplify. r (, y) y r STANDARD EQUATION OF A CIRCLE ( CENTER AT ORIGIN) The standard form of the equation of a circle with center at (0, 0) and radius r is as follows: + y = r EXAMPLE A circle with center at (0, 0) and radius 3 has equation + y = 9. E X A M P L E 1 Graphing an Equation of a Circle Draw the circle given by y = 5 º. Write the equation in standard form. y = 5 º Original equation (0, 5) + y = 5 Add to each side. In this form you can see that the graph is a circle whose center is the origin and whose radius is r = 5 = 5. ( 5, 0) (5, 0) Plot several points that are 5 units from the origin. The points (0, 5), (5, 0), (0, º5), and (º5, 0) are most convenient. Draw a circle that passes through the four points. y 5 (0, 5) 10.3 Circles 601

17 n o n p f g h i j k l m E X A M P L E Writing an Equation of a Circle STUDENT HELP HOMEWORK HELP Visit our Web site for etra eamples. INTERNET The point (1, 4) is on a circle whose center is the origin. Write the standard form of the equation of the circle. Because the point (1, 4) is on the circle, the radius of the circle must be the distance between the center and the point (1, 4). r = (1 º 0 ) + ( 4 º 0 ) = Simplify. = 1 7 Use the distance formula. Knowing that the radius is 1 7, you can use the standard form to find an equation of the circle. + y = r Standard form + y = ( 1 7 ) Substitute 1 7 for r. + y = 17 Simplify STUDENT HELP Study Tip In mathematics the term radius is used in two ways. As defined on the previous page, it is the distance from the center of a circle to a point on the circle. It can also refer to the line segment that connects the center to a point on the circle. A theorem in geometry states that a line tangent to a circle is perpendicular to the circle s radius at the point of tangency. In the diagram, AB is tangent to the circle with center C at the point of tangency B, so AB fi B C. This property of circles is used in the net eample. A B C E X A M P L E 3 Finding a Tangent Line Write an equation of the line that is tangent to the circle + y = 13 at (, 3). The slope of the radius through the point (, 3) is: m = 3 º 0 3 = º 0 Because the tangent line at (, 3) is perpendicular to this radius, its slope must be the negative reciprocal of 3, or º} }. So, an equation of the 3 tangent line is as follows. y º 3 = º} }( º ) 3 Point-slope form y º 3 = º Distributive property 3 y = º Add 3 to each side. 3 An equation of the tangent line is y = º y 3 3 (, 3) y Chapter 10 Quadratic Relations and Conic Sections

18 o y o y y o y p p p q r s t u v w FOCUS ON APPLICATIONS GOAL USING CIRCLES IN REAL LIFE The regions inside and outside the circle + y = r can be described by inequalities. Region inside circle: + y < r Region outside circle: + y > r E X A M P L E 4 Using a Circular Model THE PHAROS OF ALEXANDRIA was a lighthouse built in Egypt in about 80 B.C. One of the Seven Wonders of the World, it was said to be over 440 feet tall. It stood for nearly 1400 years. REAL LIFE OCEAN NAVIGATION The beam of a lighthouse can be seen for up to 0 miles. You are on a ship that is 10 miles east and 16 miles north of the lighthouse. a. Write an inequality to describe the region lit by the lighthouse beam. b. Can you see the lighthouse beam? a. As shown at the right the lighthouse beam can be seen from all points that satisfy this inequality: + y < 0 b. Substitute the coordinates of the ship into the inequality you wrote in part (a). + y < 0 Inequality from part (a) <? 0 Substitute for and y. y 0 (10, 16) <? 400 Simplify. 356 < 400 The inequality is true. You can see the beam from the ship. In the diagram above, the origin represents the lighthouse and the positive y-ais represents north. E X A M P L E 5 Using a Circular Model OCEAN NAVIGATION Your ship in Eample 4 is traveling due south. For how many more miles will you be able to see the beam? When the ship eits the region lit by the beam, it will be at a point on the circle + y = 0. Furthermore, its -coordinate will be 10 and its y-coordinate will be negative. Find the point (10, y) where y < 0 on the circle + y = 0. + y = 0 Equation for the boundary 10 + y = 0 Substitute 10 for. (10, 17.3) y = Solve for y. Since y < 0, y º17.3. The beam will be in view as the ship travels from (10, 16) to (10, º17.3), a distance of 16 º (º17.3) = 33.3 miles Circles 603

19 z z { { z z ~ } { { ~ } ~ z { { ~ z q r s t u v w GUIDED PRACTICE Vocabulary Check Concept Check Skill Check 1. State the definition of a circle.. LOGICAL REASONING Tell whether the following statement is always true, sometimes true, or never true: For a given circle and a given -coordinate, there are two points on the circle with that -coordinate. 3. How is the slope of a line tangent to a circle related to the slope of the radius at the point of tangency? 4. ERROR ANALYSIS A student was asked to write an equation of a circle with its center at the origin and a radius of 4. The student wrote the following equation: + y = 4 What did the student do wrong? Write the correct equation. Write the standard form of the equation of the circle that passes through the given point and whose center is the origin. 5. (4, 0) 6. (0, º) 7. (º8, 6) 8. (º5, º1) 9. (6, º9) 10. (3, 1) 11. (º5, º5) 1. (º, 4) Graph the equation. Give the radius of the circle y = y = y = y = y = y = SHOT PUT A person throws a shot put from a circle that has a diameter of 7 feet. Write the standard form of the equation of the shot put circle if the center is the origin. PRACTICE AND APPLICATIONS STUDENT HELP Etra Practice to help you master skills is on p MATCHING GRAPHS Match the equation with its graph y = y = 5. + y = y = y = y = 10 A. B. C. STUDENT HELP D. E. F. HOMEWORK HELP Eample 1: Es Eample : Es Eample 3: Es Eamples 4, 5: Es Chapter 10 Quadratic Relations and Conic Sections

20 ƒ GRAPHING Graph the equation. Give the radius of the circle y = y = y = y = y = y = y = y = y = y = y = y = 5 GRAPHING In Eercises 38 46, the equations of both circles and parabolas are given. Graph the equation y = y = y = y = y = y = º + 9y = y = y = 0 WRITING EQUATIONS Write the standard form of the equation of the circle with the given radius and whose center is the origin WRITING EQUATIONS Write the standard form of the equation of the circle that passes through the given point and whose center is the origin. 59. (0, º10) 60. (8, 0) 61. (º3, º4) 6. (º4, º1) 63. (5, º3) 64. (º6, 4) 65. (º6, 1) 66. (º1, º9) 67. (7, º4) 68. (10, ) 69. (5, 8) 70. (, º1) IRRIGATION This irrigation project in Colorado enables farmers to raise crops in the desert. Water from deep wells is pumped to sprinklers that rotate, forming circular patterns. REAL FOCUS ON APPLICATIONS LIFE FINDING TANGENT LINES The equation of a circle and a point on the circle is given. Write an equation of the line that is tangent to the circle at that point y = 10; (1, 3) 7. + y = 5; (, 1) y = 41; (º4, º5) y = 145; (1, 1) y = 65; (º8, 1) y = 40; (º, 6) y = 44; (º10, º1) y = 5 7 ; 4 1, º8 79. CRITICAL THINKING Look back at Eample 3. Find an equation of the line that is tangent to the circle at the point (, º3). Describe how the line is geometrically related to the line found in Eample Writing Describe how the equation of a circle is related to the Pythagorean theorem. Include a diagram to illustrate the relationship. 81. EARTHQUAKES Suppose an earthquake can be felt up to 80 miles from its epicenter. You are located at a point 60 miles west and 45 miles south of the epicenter. Do you feel the earthquake? If so, how many miles south would you have to travel to be out of the range of the earthquake? 8. DESERT IRRIGATION A circular field has an area of about,400,000 square yards. Write an equation that represents the boundary of the field. Let (0, 0) represent the center of the field Circles 605

21 ˆ Š Œ Ž AIR TRAFFIC CONTROLLER Air traffic controllers are responsible for making sure airplanes fly a safe distance away from one another. They also help keep flights on schedule. REAL CAREER LINK INTERNET LIFE FOCUS ON CAREERS 83. RESIZING A RING One way to resize a ring is to fit a bar into the ring, as shown. Suppose a ring that is 0 millimeters in diameter has to be resized to fit a finger 16 millimeters in diameter. What is the length of the bar that should be inserted in order to make the ring fit the finger? (Hint: Write an equation of the ring, assuming it is centered at the origin. Determine what the y-coordinate of the bar must be and then substitute this coordinate into the equation to find.) 84. LIFEGUARD You are a lifeguard at a pond. The pond is a circle with a diameter of 360 feet. You want to rope off a section of the pond for swimming. If you want the rope to form a chord of the circle and have a maimum distance of 45 feet from shore, approimately how long will you need the rope to be? 85. PHYSICAL THERAPY A tilt-board is a physical therapy device that a person rocks back and forth on. Suppose the ends of a tilt-board are part of a circle with a radius of 30 inches. If the tilt-board has a depth of 6 inches, how wide is it? Source: Steps to Follow bar finger (, y) (, y) 30 in. 6 in. Test Preparation AIR TRAFFIC CONTROL In Eercises 86 and 87, use the following information. An air traffic control tower can detect airplanes up to 50 miles away. You are in an airplane 4 miles east and 43 miles south of the control tower. 86. Write an inequality that describes the region in which planes can be detected by the control tower. Can the control tower detect your plane on its radar? 87. Suppose a jet is 35 miles west and 66 miles north of the control tower and is traveling due south at a speed of 500 miles per hour. After how many minutes will the jet appear on the control tower s radar? 88. MULTIPLE CHOICE What is the equation of the line that is tangent to the circle + y = 53 at the point (7, )? A y = º B y = º C y = 7 º 4 5 D y = º 7 º 4 5 Challenge 89. MULTIPLE CHOICE Suppose a signal from a television transmitter tower can be received up to 150 miles away. The following points represent the locations of houses near the transmitter tower with the origin representing the tower. Which point is not within the range of the tower? A (10, 0) B (40, 140) C (105, 10) D (10, 148) 90. ESTIMATING AREA The segment of a circle is the region bounded by a chord and an arc. Estimate the area of the shaded segment by finding the area of ABC and the area of ABD, given that AD Æ and BD Æ are tangent to the circle. A(3, 4) C D B(3, 4) 606 Chapter 10 Quadratic Relations and Conic Sections

22 ˆ Š Œ Ž MIXED REVIEW SOLVING SYSTEMS Solve the system using any algebraic method. (Review 3.) 91. º 9y = º y = º 3y = 6 º 5y = y = º49 º7 + 4y = º 5y = º + 5y = º9 + 4y = 15 + y = 1 4 º 9y = y = 5 COMPOSITION OF FUNCTIONS Find ƒ(g()) and g(ƒ()). (Review 7.3) 97. ƒ() = + 1 and g() = 98. ƒ() = and g() = º ƒ() = º º 1 and g() = ƒ() = º 7 and g() = GRAPHING FUNCTIONS Graph the function. (Review 8.1) 101. y = y = º y = 4 3 º 1 º y = 3 º y = + 3 º y = BABYSITTING In Eercises 107 and 108, use the following information. In June you babysit 35 hours for the Johnsons and 5 hours for the Martins. In July you babysit 11 hours for the Johnsons and 40 hours for the Martins. In August you babysit 95 hours for the Johnsons and 63 hours for the Martins. (Review 4.1) 107. Use a matri to organize the information You charge $6 per hour for babysitting. Using your matri from Eercise 107, write a matri that shows how much you earned over the summer vacation. QUIZ 1 Self-Test for Lessons Find the distance between the two points. Then find the midpoint of the line segment joining the two points. (Lesson 10.1) 1. (0, 0), (8, 6). (3, 3), (º3, º3) 3. (º, 7), (4, º10) 4. (3, º7), (º5, º9) 5. (8, 6), (º4, 4) 6. (º1, º13), (11, 15) Draw the parabola. Identify the focus and directri. (Lesson 10.) 7. y = y = 9. º = 5y 10. º4y = = 7y 1. 1 = y y = º º 1y = 0 Write the standard form of the equation of the circle that passes through the given point and whose center is the origin. (Lesson 10.3) 15. (0, 3) 16. (º5, 0) 17. (4, 7) 18. (º, º5) 19. (º1, 9) 0. (6, º3) 1. (6, º6). (º7, 8) 3. RADIO SIGNALS The signals of a radio station can be received up to 65 miles away. Your house is 35 miles east and 56 miles south of the radio station. Can you receive the radio station s signals? Eplain. (Lesson 10.3) 10.3 Circles 607

23 ˆ Š Œ Ž ACTIVITY 10.3 Using Technology Graphing Circles Graphing Calculator Activity for use with Lesson 10.3 When you use a graphing calculator to draw a circle, you need to remember two things. First, most graphing calculators cannot directly graph equations such as + y = 36 because they are not functions. Second, to obtain a graph with true perspective (in which a circle looks like a circle) you must use a square setting. EXAMPLE Use a graphing calculator to draw the graph of + y = Begin by solving the equation for y. + y = 36 y = 36 º y = ± 3 6 º Enter the two equations into the graphing calculator. Y1= (36-X ) Y=- (36-X ) Y3= Y4= Y5= Y6= Y7= STUDENT HELP KEYSTROKE HELP See keystrokes for several models of calculators at INTERNET Net set the viewing window so that it has a square setting. On some graphing calculators you can select a square setting, such as ZSquare. On a graphing calculator whose viewing window s height is two thirds its width, you can obtain a square setting by choosing maimum and minimum values that satisfy this equation: RANGE Xmin=-1 Xma=1 Xscl=1 Ymin=-8 Yma=8 Yscl=1 Y ma º Ymin = Xma º Xmin 3 3 The graph is shown at the right. (Some calculators may not connect the ends of the two graphs.) EXERCISES Use a graphing calculator to graph the equation. Write the setting of the viewing window that you used and verify that it is a square setting y = y = y = y = y = y = y = y = y = Chapter 10 Quadratic Relations and Conic Sections

24 š š 10.4 Ellipses What you should learn GOAL 1 Graph and write equations of ellipses. GOAL Use ellipses in real-life situations, such as modeling the orbit of Mars in Eample 4. Why you should learn it To solve real-life problems, such as finding the area of an elliptical Australian football field in Es REAL LIFE GOAL 1 GRAPHING AND WRITING EQUATIONS OF ELLIPSES An ellipse is the set of all points P such that the sum of the distances between P and two distinct fied points, called the foci, is a constant. focus The line through the foci intersects the ellipse at two points, the vertices. The line segment d 1 d constant joining the vertices is the major ais, and its midpoint is the center of the ellipse. The line perpendicular to the major ais at the center intersects the ellipse at two points called the co-vertices. The line segment that joins these points is the minor ais of the ellipse. The two types of ellipses we will discuss are those with a horizontal major ais and those with a vertical major ais. verte: ( a, 0) major ais focus: ( c, 0) co-verte: (0, b) focus: (c, 0) co-verte: (0, b) verte: (a, 0) minor ais d 1 focus co-verte: ( b, 0) major ais d P verte: (0, a) focus: (0, c) focus: (0, c) verte: (0, a) co-verte: (b, 0) minor ais Ellipse with horizontal major ais y + } } = 1 a b Ellipse with vertical major ais y + } } = 1 b a CHARACTERISTICS OF AN ELLIPSE ( CENTER AT ORIGIN) The standard form of the equation of an ellipse with center at (0, 0) and major and minor aes of lengths a and b, where a > b > 0, is as follows. EQUATION MAJOR AXIS VERTICES CO-VERTICES y + = 1 Horizontal (±a, 0) (0, ±b) a b y + = 1 Vertical (0, ±a) (±b, 0) b a The foci of the ellipse lie on the major ais, c units from the center where c = a º b Ellipses 609

25 š š š œ ž Ÿ E X A M P L E 1 Graphing an Equation of an Ellipse STUDENT HELP HOMEWORK HELP Visit our Web site for etra eamples. INTERNET Draw the ellipse given by y = 144. Identify the foci. First rewrite the equation in standard form y = 1 44 Divide each side by y = Simplify. Because the denominator of the -term is greater than that of the y -term, the major ais is horizontal. So, a = 4 and b = 3. Plot the vertices and co-vertices. Then draw the ellipse that passes through these four points. The foci are at (c, 0) and (ºc, 0). To find the value of c, use the equation c = a º b. c = 4 º 3 = 16 º 9 = 7 c = 7 The foci are at ( 7, 0) and (º 7, 0). E X A M P L E Writing Equations of Ellipses Write an equation of the ellipse with the given characteristics and center at (0, 0). a. Verte: (0, 7) b. Verte: (º4, 0) Co-verte: (º6, 0) Focus: (, 0) ( 4, 0) (0, 3) (4, 0) (0, 3) In each case, you may wish to draw the ellipse so that you have something to check your final equation against. a. Because the verte is on the y-ais and the co-verte is on the -ais, the major ais is vertical with a = 7 and b = 6. y An equation is + = co-verte ( 6, 0) verte (0, 7) verte (0, 7) co-verte (6, 0) b. Because the verte and focus are points on a horizontal line, the major ais is horizontal with a = 4 and c =. To find b, use the equation c = a º b. = 4 º b b = 16 º 4 = 1 verte ( 4, 0) focus (, 0) focus (, 0) verte (4, 0) b = 3 y An equation is + = Chapter 10 Quadratic Relations and Conic Sections

26 œ ž Ÿ GOAL USING ELLIPSES IN REAL LIFE Both man-made objects, such as The Ellipse at the White House, and natural phenomena, such as the orbits of planets, involve ellipses. REAL LIFE E X A M P L E 3 Finding the Area of an Ellipse Landscaping A portion of the White House lawn is called The Ellipse. It is 1060 feet long and 890 feet wide. a. Write an equation of The Ellipse. b. The area of an ellipse is A = πab. What is the area of The Ellipse at the White House? Old Eecutive Office Building White House Treasury Department a. The major ais is horizontal with a = = 530 and b = 89 0 = 445. The Ellipse An equation is y = b. The area is A = π(530)(445) 741,000 square feet. REAL LIFE Astronomy E X A M P L E 4 Modeling with an Ellipse In its elliptical orbit, Mercury ranges from million kilometers to million kilometers from the center of the sun. The center of the sun is a focus of the orbit. Write an equation of the orbit. Using the diagram shown, you can write a system of linear equations involving a and c. a º c = a + c = Adding the two equations gives a = 115.9, so a = Substituting this a-value into the second equation gives c = 69.86, so c = From the relationship c = a º b, you can conclude the following: b = a º c = ( ) º ( ) An equation of the elliptical orbit is y + (57.95) = 1 where and y are (56.71) in millions of kilometers a ª c sun a 10.4 Ellipses 611

27 µ ³ µ «± ² GUIDED PRACTICE Vocabulary Check Concept Check Skill Check 1. Complete each statement using the ellipse shown. a. The points (º5, 0) and (5, 0) are called the?. b. The points (0, º4) and (0, 4) are called the?. c. The points (º3, 0) and (3, 0) are called the?. d. The segment with endpoints (º5, 0) and (5, 0) is called the?.. How can you tell from the equation of an ellipse whether the major ais is horizontal or vertical? 3. Eplain how to find the foci of an ellipse given the coordinates of its vertices and co-vertices. 4. ERROR ANALYSIS A student was asked to write an equation of the ellipse shown at the right. The student wrote the equation + y = What did the student do wrong? What is the correct equation? Write an equation of the ellipse with the given characteristics and center at (0, 0). E Verte: (0, 5) 6. Verte: (9, 0) 7. Verte: (º7, 0) Co-verte: (º4, 0) Co-verte: (0, ) Focus: (º 1 0, 0) 8. Verte: (0, 13) 9. Co-verte: ( 9 1, 0) 10. Co-verte: (0, 3 3 ) Focus: (0, º5) Focus: (0, 3) Focus: (4, 0) 1 y 1 E. 4 Draw the ellipse. y = 1 1. y + = y = y = y = y = GARDEN An elliptical garden is 10 feet long and 6 feet wide. Write an equation for the garden. Then graph the equation. Label the vertices, co-vertices, and foci. Assume that the major ais of the garden is horizontal. PRACTICE AND APPLICATIONS STUDENT HELP Etra Practice to help you master skills is on p IDENTIFYING PARTS Write the equation in standard form (if not already). Then identify the vertices, co-vertices, and foci of the ellipse. y = y + = y = y = y. + = y 3. + = y = y = y = y = y = y = Chapter 10 Quadratic Relations and Conic Sections

28 ¹ º» ¼ ½ STUDENT HELP HOMEWORK HELP Eample 1: Es Eample : Es Eamples 3, 4: Es GRAPHING Graph the equation. Then identify the vertices, co-vertices, and foci of the ellipse. y = y + = y = y = y = y = y 3. + = y = y = y = y = y 1 + = GRAPHING In Eercises 4 50, the equations of parabolas, circles, and ellipses are given. Graph the equation y = y = y + = = 144y y = y = y + = y = y = 45 WRITING EQUATIONS Write an equation of the ellipse with the given characteristics and center at (0, 0). 51. Verte: (0, 6) 5. Verte: (0, 6) 53. Verte: (º4, 0) Co-verte: (5, 0) Co-verte: (º, 0) Co-verte: (0, 3) 54. Verte: (0, º7) 55. Verte: (9, 0) 56. Verte: (10, 0) Co-verte: (º1, 0) Co-verte: (0, º8) Co-verte: (0, 4) 57. Verte: (0, 7) 58. Verte: (º5, 0) 59. Verte: (0, 8) Focus: (0, 3) Focus: ( 6, 0) Focus: (0, º4 3 ) 60. Verte: (15, 0) 61. Verte: (5, 0) 6. Verte: (0, º30) Focus: (1, 0) Focus: (º3, 0) Focus: (0, 0) 63. Co-verte: ( 5 5, 0) 64. Co-verte: (0, º 3 ) 65. Co-verte: (º 1 0, 0) Focus: (0, º3) Focus: (º1, 0) Focus: (0, 9) 66. Co-verte: (0, º3 3 ) 67. Co-verte: (5 1 1, 0) 68. Co-verte: (0, º 7 7 ) Focus: (3, 0) Focus: (0, º7) Focus: (º, 0) WHISPERING GALLERY In Eercises 69 71, use the following information. Statuary Hall is an elliptical room in the United States Capitol in Washington, D.C. The room is also called the Whispering Gallery because a person standing at one focus of the room can hear even a whisper spoken by a person standing at the other focus. This occurs because any sound that is emitted from one focus of an ellipse will reflect off the side of the ellipse to the other focus. Statuary Hall is 46 feet wide and 97 feet long. 69. Find an equation that models the shape of the room. 70. How far apart are the two foci? 71. What is the area of the floor of the room? 10.4 Ellipses 613

29 ¹ º» ¼ ½ 7. SPACE EXPLORATION The first artificial satellite to orbit Earth was Sputnik I, launched by the Soviet Union in The orbit was an ellipse with Earth s center as one focus. The orbit s highest point above Earth s surface was 583 miles, and its lowest point was 13 miles. Find an equation of the orbit. (Use 4000 miles as the radius of Earth.) Graph your equation. Test Preparation Challenge AUSTRALIAN FOOTBALL In Eercises 73 75, use the information below. Australian football is played on an elliptical field. The official rules state that the field must be between 135 and 185 meters long and between 110 and 155 meters wide. Source: The Australian News Network 73. Write an equation for the largest allowable playing field. 74. Write an equation for the smallest allowable playing field. 75. Write an inequality that describes the possible areas of an Australian football field. 76. MULTI-STEP PROBLEM A tour boat travels between two islands that are 1 miles apart. For a trip between the islands, there is enough fuel for a 0-mile tour. a. Writing The region in which the boat can travel is bounded by an ellipse. Eplain why this is so. b. Let (0, 0) represent the center of the ellipse. Find the coordinates of each island. c. Suppose the boat travels from one island, straight past the other island to the verte of the ellipse, and back to the second island. How many miles does the boat travel? Use your answer to find the coordinates of the verte. d. Use your answers to parts (b) and (c) to write an equation for the ellipse that bounds the region the boat can travel in. 77. LOGICAL REASONING Show that c = a º b for any ellipse given by the y equation + a = 1 with foci at (c, 0) and (ºc, 0). b Island 1 Island MIXED REVIEW RATIONAL EXPONENTS Evaluate the epression without using a calculator. (Review 7.1) /3 79. º8 5/ / º/ / / º/ /5 INVERSE VARIATION The variables and y vary inversely. Use the given values to write an equation relating and y. (Review 9.1) 86. = 3, y = º 87. = 4, y = = 5, y = = 8, y = = 9, y = 91. = 0.5, y = 4 GRAPHING Graph the function. State the domain and range. (Review 9. for 10.5) 9. ƒ() = ƒ() = º ƒ() = ƒ() = ƒ() = 97. ƒ() = º Chapter 10 Quadratic Relations and Conic Sections

30 À ¾ ¹ º» ¼ ½ 10.5 Hyperbolas What you should learn GOAL 1 Graph and write equations of hyperbolas. GOAL Use hyperbolas to solve real-life problems, such as modeling a sundial in Es Why you should learn it To model real-life objects, such as a sculpture in Eample 4. REAL LIFE GOAL 1 GRAPHING AND WRITING EQUATIONS OF HYPERBOLAS The definition of a hyperbola is similar to that of an ellipse. For an ellipse, recall that the sum of the distances between a point on the ellipse and the two foci is constant. For a hyperbola, the difference is constant. A hyperbola is the set of all points P such that the difference of the distances P d from P to two fied points, called the d 1 foci, is constant. The line through the foci focus focus intersects the hyperbola at two points, the d d 1 constant vertices. The line segment joining the vertices is the transverse ais, and its midpoint is the center of the hyperbola. A hyperbola has two branches and two asymptotes. The asymptotes contain the diagonals of a rectangle centered at the hyperbola s center, as shown below. (0, b) focus: (0, c) verte: (0, a) verte: ( a, 0) focus: ( c, 0) verte: (a, 0) focus: (c, 0) ( b, 0) (b, 0) transverse ais transverse ais (0, b) focus: (0, c) verte: (0, a) Hyperbola with horizontal transverse ais y } º } = 1 a b Hyperbola with vertical transverse ais y º } } = 1 a b CHARACTERISTICS OF A HYPERBOLA ( CENTER AT ORIGIN) The standard form of the equation of a hyperbola with center at (0, 0) is as follows. EQUATION TRANSVERSE AXIS ASYMPTOTES VERTICES y º = 1 Horizontal y = ± b (±a, 0) a b a y º a = 1 Vertical y = ± (0, ±a) a b b The foci of the hyperbola lie on the transverse ais, c units from the center where c = a + b Hyperbolas 615

31 É À É Ë À Ê À Á Â Ã Ä Å Æ Ç È E X A M P L E 1 Graphing an Equation of a Hyperbola Draw the hyperbola given by 4 º 9y = 36. First rewrite the equation in standard form. 4 º 9y = 36 Write original equation º 9 y 3 = 3 6 Divide each side by º y = Simplify. Note from the equation that a = 9 and b = 4, so a = 3 and b =. Because the -term is positive, the transverse ais is horizontal and the vertices are at (º3, 0) and (3, 0). To draw the hyperbola, first draw a rectangle that is centered at the origin, a = 6 units wide and b = 4 units high. Then show the asymptotes by drawing the lines that pass through opposite corners of the rectangle. Finally, draw the hyperbola. (0, ) (0, ) ( 3, 0) (3, 0) ( 3, 0) (3, 0) (0, ) (0, ) E X A M P L E Writing an Equation of a Hyperbola Write an equation of the hyperbola with foci at (0, º3) and (0, 3) and vertices at (0, º) and (0, ). The transverse ais is vertical because the foci and vertices lie on the y-ais. The center is the origin because the foci and the vertices are equidistant from the origin. Since the foci are each 3 units from the center, c = 3. Similarly, because the vertices are each units from the center, a =. You can use these values of a and c to find b. b = c º a b = 3 º = 9 º 4 = 5 5, 0 (0, ) (0, ) (0, 3) (0, 3) 5, 0 b = 5 Because the transverse ais is vertical, the standard form of the equation is as follows. y º = 1 Substitute for a and 5 for b. y º = Simplify. 616 Chapter 10 Quadratic Relations and Conic Sections

32 Ì À Á Â Ã Ä Å Æ Ç È FOCUS ON APPLICATIONS GOAL USING HYPERBOLAS IN REAL LIFE E X A M P L E 3 Using a Real-Life Hyperbola PHOTOGRAPHY A hyperbolic mirror can be used to take panoramic photographs. A camera is pointed toward the verte of the mirror and is positioned so that the lens is at one focus of the mirror. An equation for the cross section of the mirror y is º = 1 where and y are measured in inches. How far from the mirror is the lens? PANORAMIC CAMERAS A panoramic photograph taken with the camera shown above gives a 360 view of a scene. REAL APPLICATION LINK INTERNET LIFE Notice from the equation that a = 16 and b = 9, so a = 4 and b = 3. Use these values and the equation c = a + b to find the value of c. c = a + b c = = 5 c = 5 Solve for c. Equation relating a, b, and c Substitute for a and b and simplify. Since a = 4 and c = 5, the vertices are at (0, º4) and (0, 4) and the foci are at (0, º5) and (0, 5). The camera is below the mirror, so the lens is at (0, º5) and the verte of the mirror is at (0, 4). The distance between these points is 4 º (º5) = 9. The lens is 9 inches from the mirror. REAL LIFE E X A M P L E 4 Modeling with a Hyperbola Sculpture The diagram at the right shows the hyperbolic cross section of a sculpture located at the Fermi National Accelerator Laboratory in Batavia, Illinois. (, 13) (, 13) a. Write an equation that models the curved sides of the sculpture. b. At a height of 5 feet, how wide is the sculpture? (Each unit in the coordinate plane represents 1 foot.) ( 1, 0) (1, 0) Ê a. From the diagram you can see that the transverse ais is horizontal and a = 1. So the equation has this form: 1 º y b = 1 (, 13) (, 13) Because the hyperbola passes through the point (, 13), you can substitute = and y = 13 into the equation and solve for b. When you do this, you obtain b 7.5. An equation of the hyperbola is º 1 y (7.5) = 1. b. At a height of 5 feet above the ground, y = º8. To find the width of the sculpture, substitute this value into the equation and solve for. You get At a height of 5 feet, the width is.9 feet Hyperbolas 617

33 Ö Ø Ö Ø Õ Õ Ö Ø Ö Ø Ø Õ Õ Ø Õ Í Î Ï Ð Ñ Ò Ó Ô GUIDED PRACTICE Vocabulary Check Concept Check Skill Check 1. Complete these statements: The points (0, º) and (0, ) in the graph at the right are the? of the hyperbola. The segment joining these two points is the?.. How are the definitions of ellipse and hyperbola alike? How are they different? 3. How do the asymptotes of a hyperbola help you draw the hyperbola? Graph the equation. Identify the foci and asymptotes. y 4. º = y 5. º = (0, ) (0, ) E º y 6 4 = º 4y = y º 5 = y º 9 = 9 Write an equation of the hyperbola with the given foci and vertices. 10. Foci: (0, º5), (0, 5) 11. Foci: (º8, 0), (8, 0) Vertices: (0, º3), (0, 3) Vertices: (º7, 0), (7, 0) 1. Foci: (º 3 4, 0), ( 3 4, 0) 13. Foci: (0, º9), (0, 9) Vertices: (º5, 0), (5, 0) Vertices: (0, º3 5 ), (0, 3 5 ) 14. PHOTOGRAPHY Look back at Eample 3. Suppose a mirror has a cross section modeled by the equation º y = 1 where and y are measured in 5 9 inches. If you place a camera with its lens at the focus, how far is the lens from the verte of the mirror? PRACTICE AND APPLICATIONS STUDENT HELP Etra Practice to help you master skills is on p MATCHING Match the equation with its graph. 15. º y = y º = A. B. y 17. º = º y = STUDENT HELP C. D. HOMEWORK HELP Eample 1: Es Eample : Es Eample 3: E. 67 Eample 4: Es Chapter 10 Quadratic Relations and Conic Sections

34 Í Î Ï Ð Ñ Ò Ó Ô STANDARD FORM Write the equation of the hyperbola in standard form º 9y = y º 81 = y º 4 = 9. 16y º = 0 3. y º = 4 4. º 4 y = IDENTIFYING PARTS Identify the vertices and foci of the hyperbola. 5. y º = y 6. º 4 9 = 1 7. º y = y º 81 = y º 4 = º 10y = 360 GRAPHING Graph the equation. Identify the foci and asymptotes. y 31. º = y º = º y 3 6 = 1 y 35. º = y 33. º = y 36. º 6 4 = 1 STUDENT HELP HOMEWORK HELP Visit our Web site for help with problem solving in E. 49. INTERNET y º = y 38. º = º 9 y = y 40. º = º 81y = º 9y = 5 GRAPHING HYPERBOLAS Use a graphing calculator to graph the equation. Tell what two equations you entered into the calculator. y 43. º = y 46. º = y 44. º = y 45. º = y 47. º = º 8.5y = CRITICAL THINKING Suppose you tried to graph an equation of a hyperbola on a graphing calculator. You enter one function correctly, but you forget to enter the other function. Sketch what your graph might look like if the transverse ais is horizontal. Then sketch what your graph might look like if the transverse ais is vertical. GRAPHING CONIC SECTIONS In Eercises 50 55, the equations of parabolas, circles, ellipses, and hyperbolas are given. Graph the equation. y 50. º = y = y º = y = 15y = y = 16 WRITING EQUATIONS Write an equation of the hyperbola with the given foci and vertices. 56. Foci: (0, º13), (0, 13) 57. Foci: (º8, 0), (8, 0) Vertices: (0, º5), (0, 5) Vertices: (º6, 0), (6, 0) 58. Foci: (º4, 0), (4, 0) 59. Foci: (º6, 0), (6, 0) Vertices: (º1, 0), (1, 0) Vertices: (º5, 0), (5, 0) 60. Foci: (0, º7), (0, 7) 61. Foci: (0, º9), (0, 9) Vertices: (0, º3), (0, 3) Vertices: (0, º8), (0, 8) 6. Foci: (º8, 0), (8, 0) 63. Foci: (0, º5 6 ), (0, 5 6 ) Vertices: (º4 3, 0), (4 3, 0) Vertices: (0, º4), (0, 4) 10.5 Hyperbolas 619

35 á â Ù Ú Û Ü Ý Þ ß à SUNDIAL The Richard D. Swensen sundial, located at the University of Wisconsin River Falls, gives the correct time to the minute. REAL APPLICATION LINK INTERNET FOCUS ON APPLICATIONS LIFE Test Preparation Challenge EXTRA CHALLENGE SUNDIAL In Eercises 64 66, use the following information. The sundial at the left was designed by Professor John Shepherd. The shadow of the gnomon traces a hyperbola throughout the day. Aluminum rods form the hyperbolas traced on the summer solstice, June 1, and the winter solstice, December One focus of the summer solstice hyperbola is 07 inches above the ground. The verte of the aluminum branch is 66 inches above the ground. If the -ais is 355 inches above the ground and the center of the hyperbola is at the origin, write an equation for the summer solstice hyperbola. 65. One focus of the winter solstice hyperbola is 419 inches above the ground. The verte of the aluminum branch is 387 inches above the ground and the center of the hyperbola is at the origin. If the -ais is 355 inches above the ground, write an equation for the winter solstice hyperbola. 66. Use your equations from Eercises 64 and 65 to draw the lower branch of the summer solstice hyperbola and the upper branch of the winter solstice hyperbola. 67. AERONAUTICS When an airplane travels faster than the speed of sound, the sound waves form a cone behind the airplane. If the airplane is flying parallel to the ground, the sound waves intersect the ground in a hyperbola with the airplane directly above its center. A sonic boom is heard along the hyperbola. If you hear a sonic boom that is audible along a hyperbola with the equation º y = 1 where and y are measured in miles, what is the shortest horizontal distance you could be to the airplane? 68. MULTI-STEP PROBLEM Suppose you are making a ring out of clay for a necklace. If you have a fied volume of clay and you want the ring to have a certain thickness, the area of the ring becomes fied. However, you can still vary the inner radius and the outer radius y. a. Suppose you want to make a ring with an area of square inches. Write an equation relating and y. b. Find three coordinate pairs (, y) that satisfy the relationship from part (a). Then find the width of the ring, y º, for each coordinate pair. c. Writing How does the width of the ring, y º, change as and y both increase? Eplain why this makes sense. 69. Use the diagram at the right to show that d º d 1 = a. 70. LOCATING AN EXPLOSION Two microphones, 1 mile apart, record an eplosion. Microphone A receives the sound seconds after Microphone B. Is this enough information to decide where the sound came from? Use the fact that sound travels at 1100 feet per second. ( c, 0) shock wave ground y (, y) d d 1 ( a, 0) (a, 0) (c, 0) E. 69 fied 60 Chapter 10 Quadratic Relations and Conic Sections

36 ã ä å æ ç è é ê MIXED REVIEW GRAPHING FUNCTIONS Graph the function. (Review.8, 5.1 for 10.6) 71. y = y = º y = º º 6 º y = 3( º 1) y = º( º 3) º y = 1 ( + 4) + 5 WRITING FUNCTIONS Write a polynomial function of least degree that has real coefficients, the given zeros, and a leading coefficient of 1. (Review 6.7) 77. 3, 1, 78. º7, º1, , º, 80. º6, 4, 81. 5, i, ºi 8. 3, º3, i EVALUATING LOGARITHMIC EXPRESSIONS Evaluate the epression without using a calculator. (Review 8.4) 83. log 10, log log log log log log log ,000, TEST SCORES Find the mean, median, mode(s), and range of the following set of test scores. (Review 7.7) 63, 67, 7, 75, 77, 78, 81, 81, 85, 86, 89, 89, 91, 9, 99 QUIZ Self-Test for Lessons 10.4 and 10.5 Write an equation of the ellipse with the given characteristics and center at (0, 0). (Lesson 10.4) 1. Verte: (0, 7). Verte: (º6, 0) 3. Verte: (º10, 0) Co-verte: (º3, 0) Co-verte: (0, º1) Focus: (6, 0) 4. Verte: (0, 5) 5. Co-verte: (0, 3 ) 6. Co-verte: (º9, 0) Focus: (0, 1 7 ) Focus: (º 3, 0) Focus: (0, 4) Graph the equation. Identify the vertices, co-vertices, and foci. (Lesson 10.4) 7. y + = y = y = 36 Write an equation of the hyperbola with the given characteristics. (Lesson 10.5) 10. Foci: (0, º8), (0, 8) 11. Foci: (º3, 0), (3, 0) Vertices: (0, º5), (0, 5) Vertices: (º1, 0), (1, 0) 1. Foci: (º6, 0), (6, 0) 13. Foci: (0, º 5 ), (0, 5 ) Vertices: (º4, 0), (4, 0) Vertices: (0, º4), (0, 4) Graph the equation. Identify the vertices, foci, and asymptotes. (Lesson 10.5) y 14. º = y º 0 = º 4y = SPACE EXPLORATION Suppose a satellite s orbit is an ellipse with Earth s center at one focus. If the satellite s least distance from Earth s surface is 150 miles and its greatest distance from Earth s surface is 600 miles, write an equation for the ellipse. (Use 4000 miles as Earth s radius.) (Lesson 10.4) 10.5 Hyperbolas 61

37 ã ä å æ ç è é ê ACTIVITY 10.6 Developing Concepts Eploring Conic Sections Group Activity for use with Lesson 10.6 GROUP ACTIVITY Work with a partner. MATERIALS flashlight graph paper pencil QUESTION How do a plane and a double-napped cone intersect to form different conic sections? EXPLORING THE CONCEPT The reason that parabolas, circles, ellipses, and hyperbolas are called conics or conic sections is that each can be formed by the intersection of a plane and a double-napped cone, as shown below. Circle Ellipse Parabola Hyperbola The beam of light from a flashlight is a cone. When the light hits a flat surface such as a wall, the edge of the beam of light forms a conic section. Work in a group to find an equation of a conic formed by a flashlight beam On a piece of graph paper, draw - and y-aes to make a coordinate plane. Tape the paper to a wall. Aim a flashlight perpendicular to the paper so that the light forms a circle. Move the flashlight so that the circle is centered on the origin of the coordinate plane. Holding the flashlight very still, trace the circle on the graph paper. Find the radius of the circle and use it to write the standard form of the equation of the circle. Aim the flashlight at the paper to form an ellipse with a vertical major ais and center at the origin. Trace the ellipse and find the standard form of its equation. DRAWING CONCLUSIONS 1. Compare the equation of your circle with the equations found by other groups. Are your equations all the same? Why or why not?. Compare the equation of your ellipse with the equations found by other groups. Are your equations all the same? Why or why not? 6 Chapter 10 Quadratic Relations and Conic Sections

38 ë í ë ì ã ä å æ ç è é ê 10.6 Graphing and Classifying Conics What you should learn GOAL 1 Write and graph an equation of a parabola with its verte at (h, k) and an equation of a circle, ellipse, or hyperbola with its center at (h, k). GOAL Classify a conic using its equation, as applied in Eample 8. Why you should learn it To model real-life situations involving more than one conic, such as the circles that an ice skater uses to practice figure eights in E. 64. REAL LIFE GOAL 1 WRITING AND GRAPHING EQUATIONS OF CONICS Parabolas, circles, ellipses, and hyperbolas are all curves that are formed by the intersection of a plane and a double-napped cone. Therefore, these shapes are called conic sections or simply conics. In previous lessons you studied equations of parabolas with vertices at the origin and equations of circles, ellipses, and hyperbolas with centers at the origin. In this lesson you will study equations of conics that have been translated in the coordinate plane. STANDARD FORM OF EQUATIONS OF TRANSLATED CONICS In the following equations the point (h, k) is the verte of the parabola and the center of the other conics. CIRCLE ( º h) + (y º k) = r Horizontal ais Vertical ais PARABOLA (y º k) = 4p( º h) ( º h) = 4p(y º k) ELLIPSE ( º h) (y º k) a + ( º h) (y º k) = 1 + = 1 b b a HYPERBOLA ( º h) (y º k) a º (y º k) = 1 º ( º h) b = 1 b a E X A M P L E 1 Writing an Equation of a Translated Parabola Write an equation of the parabola whose verte is at (º, 1) and whose focus is at (º3, 1). Choose form: Begin by sketching the parabola, as shown. Because the parabola opens to the left, it has the form ( 3, 1) where p < 0. (y º k) = 4p( º h) (, 1) Find h and k: The verte is at (º, 1), so h = º and k = 1. Find p: The distance between the verte (º, 1) and the focus (º3, 1) is p = (º 3 º ( º )) + ( 1 º 1 ) = 1 so p = 1 or p = º1. Since p < 0, p = º1. The standard form of the equation is (y º 1) = º4( + ) Graphing and Classifying Conics 63

39 ö ù ö ú û ü ø î ï ð ñ ò ó ô õ E X A M P L E Graphing the Equation of a Translated Circle Graph ( º 3) + (y + ) = 16. Compare the given equation to the standard form of the equation of a circle: ( º h) + (y º k) = r You can see that the graph is a circle with center at (h, k) = (3, º) and radius r = 4. Plot the center. Plot several points that are each 4 units from the center: (3 + 4, º) = (7, º) (3 º 4, º) = (º1, º) (3, º + 4) = (3, ) (3, º º 4) = (3, º6) Draw a circle through the points. ( 1, ) (3, ) (3, ) (3, 6) (7, ) E X A M P L E 3 Writing an Equation of a Translated Ellipse STUDENT HELP HOMEWORK HELP Visit our Web site for etra eamples. INTERNET Write an equation of the ellipse with foci at (3, 5) and (3, º1) and vertices at (3, 6) and (3, º). Plot the given points and make a rough sketch. The ellipse has a vertical major ais, so its equation is of this form: ( º h) b + (y º a k) = 1 (3, 6) (3, 5) Find the center: The center is halfway between the vertices. (h, k) = 3 + 3, 6 + (º) = (3, ) Find a: The value of a is the distance between the verte and the center. (3, ) (3, 1) a = (3 º 3 ) + ( 6 º ) = = 4 Find c: The value of c is the distance between the focus and the center. c = (3 º 3 ) + ( 5 º ) = = 3 Find b: Substitute the values of a and c into the equation b = a º c. b = 4 º 3 b = 7 b = 7 The standard form of the equation is ( º 3) + (y º ) = Chapter 10 Quadratic Relations and Conic Sections

40 ý þ ÿ E X A M P L E 4 Graphing the Equation of a Translated Hyperbola Graph (y + 1) º ( + 1) = 1. 4 The y -term is positive, so the transverse ais is vertical. Since a = 1 and b = 4, you know that a = 1 and b =. Plot the center at (h, k) = (º1, º1). Plot the vertices 1 unit above and below the center at (º1, 0) and (º1, º). Draw a rectangle that is centered at (º1, º1) and is a = units high and b = 4 units wide. Draw the asymptotes through the corners of the rectangle. ( 1, 1) ( 1, 0) ( 1, ) Draw the hyperbola so that it passes through the vertices and approaches the asymptotes. E X A M P L E 5 Using Circular Models COMMUNICATIONS A cellular phone transmission tower located 10 miles west and 5 miles north of your house has a range of 0 miles. A second tower, 5 miles east and 10 miles south of your house, has a range of 15 miles. a. Write an inequality that describes each tower s range. b. Do the two regions covered by the towers overlap? FOCUS ON APPLICATIONS a. Let the origin represent your house. The first tower is at (º10, 5) and the boundary of its range is a circle with radius 0. Substitute º10 for h, 5 for k, and 0 for r into the standard form of the equation of a circle. ( º h) + (y º k) = r Standard form of a circle ( + 10) + ( y º 5) < 400 Region inside the circle The second tower is at (5, º10). The boundary of its range is a circle with radius 15. ( º h) + (y º k) = r Standard form of a circle ( º 5) + ( y + 10) < 5 Region inside the circle CELLULAR PHONES work only when there is a transmission tower nearby to retrieve the signal. Because of the need for many towers, they are often designed to blend in with the environment. REAL LIFE b. One way to tell if the regions overlap is to graph the inequalities. You can see that the regions do overlap. You can also check whether the distance between the two towers is less than the sum of the ranges. (º 1 0 º 5 ) + ( 5 º ( º 1 0 )) <? The regions do overlap. 15 <? < 35 ( 10, 5) (5, 10) 10.6 Graphing and Classifying Conics 65

41 GOAL CLASSIFYING A CONIC FROM ITS EQUATION The equation of any conic can be written in the form A + By + Cy + D + Ey + F = 0 which is called a general second-degree equation in and y. The epression B º 4AC is called the discriminant of the equation and can be used to determine which type of conic the equation represents. CONCEPT SUMMARY CLASSIFYING CONICS If the graph of A + By + Cy + D + Ey + F = 0 is a conic, then the type of conic can be determined as follows. DISCRIMINANT B º 4AC < 0, B = 0, and A = C B º 4AC < 0 and either B 0 or A C B º 4AC = 0 B º 4AC > 0 TYPE OF CONIC Circle Ellipse Parabola Hyperbola If B = 0, each ais of the conic is horizontal or vertical. If B 0, the aes are neither horizontal nor vertical. E X A M P L E 6 Classifying a Conic a. Classify the conic given by + y º 4 º 4 = 0. b. Graph the equation in part (a). STUDENT HELP Look Back For help with completing the square, see p. 8. a. Since A =, B = 0, and C = 1, the value of the discriminant is as follows: B º 4AC = 0 º 4()(1) = º8 Because B º 4AC < 0 and A C, the graph is an ellipse. b. To graph the ellipse, first complete the square as follows. + y º 4 º 4 = 0 ( º 4) + y = 4 ( º ) + y = 4 ( º +? ) + y = 4 + (? ) ( º + 1) + y = 4 + (1) ( º 1) + y = 6 ( º 1) + y = , 0 1, 6 (1, 0) 1, 6 By comparing this equation to ( º b h) + (y º a k) = 1, you can see that h = 1, k = 0, a = 6, and b = 3. Use these facts to draw the ellipse. 1 3, 0 66 Chapter 10 Quadratic Relations and Conic Sections

42 ! E X A M P L E 7 Classifying a Conic a. Classify the conic given by 4 º 9y + 3 º 144y º 548 = 0. b. Graph the equation in part (a). a. Since A = 4, B = 0, and C = º9, the value of the discriminant is as follows: B º 4AC = 0 º 4(4)(º9) = 144 Because B º 4AC > 0, the graph is a hyperbola. b. To graph the hyperbola, first complete the square as follows. 4 º 9y + 3 º 144y º 548 = 0 (4 + 3) º (9y + 144y) = 548 4( + 8 +? ) º 9(y + 16y +? ) = (? ) º 9(? ) 4( ) º 9(y + 16y + 64) = (16) º 9(64) 4( + 4) º 9(y + 8) = 36 ( + 4) (y + 8) 3 º = 1 By comparing this equation to ( a h) º (y º b k) = 1, you can see that h = º4, k = º8, a = 3, and b =. To draw the hyperbola, plot the center at (h, k) = (º4, º8) and the vertices at (º7, º8) and (º1, º8). Draw a rectangle a = 6 units wide and b = 4 units high and centered at (º4, º8). Draw the asymptotes through the corners of the rectangle. Then draw the hyperbola so that it passes through the vertices and approaches the asymptotes. ( 7, 8)! ( 4, 8) ( 1, 8) REAL LIFE E X A M P L E 8 Classifying Conics in Real Life Astronomy The diagram at the right shows the mirrors in a Cassegrain telescope. The equations of the two mirrors are given below. Classify each mirror as parabolic, elliptical, or hyperbolic. eyepiece mirror B a. Mirror A: y º 7 º 450 = 0 b. Mirror B: 88.4 º 49.7y º 4390 = 0 mirror A star EQUATION B º 4AC TYPE OF MIRROR a. y º 7 º 450 = 0 0 º 4(0)(1) = 0 Parabolic b º 49.7y º 4390 = 0 0 º 4(88.4)(º49.7) > 0 Hyperbolic 10.6 Graphing and Classifying Conics 67

43 GUIDED PRACTICE Vocabulary Check Concept Check Skill Check 1. Eplain why circles, ellipses, parabolas, and hyperbolas are called conic sections.. How are the graphs of + y = 5 and ( º 1) + (y + ) = 5 alike? How are they different? 3. How can the discriminant B º 4AC be used to classify the graph of A + By + Cy + D + Ey + F = 0? Write an equation for the conic section. 4. Circle with center at (4, º1) and radius 7 5. Ellipse with foci at (, º4) and (5, º4) and vertices at (º1, º4) and (8, º4) 6. Parabola with verte at (3, º) and focus at (3, º4) 7. Hyperbola with foci at (5, ) and (5, º6) and vertices at (5, 0) and (5, º4) Classify the conic section º 4y + 4 = º 5y º 6 + y º = y + 7 º 4y º 8 = º5 º y + º 3y + 1 = 0 1. COMMUNICATIONS Look back at Eample 5. Suppose there is a tower 5 miles east and 30 miles north of your house with a range of 5 miles. Does the region covered by this tower overlap the regions covered by the two towers in Eample 5? Illustrate your answer with a graph. PRACTICE AND APPLICATIONS STUDENT HELP Etra Practice to help you master skills is on p STUDENT HELP HOMEWORK HELP Eamples 1, 3: Es Eamples, 4: Es. 1 8 Eample 5: Es. 63, 64 Eamples 6, 7: Es. 9 6 Eample 8: Es WRITING EQUATIONS Write an equation for the conic section. 13. Circle with center at (9, 3) and radius Circle with center at (º4, ) and radius Parabola with verte at (1, º) and focus at (1, 1) 16. Parabola with verte at (º3, 1) and directri = º8 17. Ellipse with vertices at (, º3) and (, 6) and foci at (, 0) and (, 3) 18. Ellipse with vertices at (º, ) and (4, ) and co-vertices at (1, 1) and (1, 3) 19. Hyperbola with vertices at (5, º4) and (5, 4) and foci at (5, º6) and (5, 6) 0. Hyperbola with vertices at (º4, ) and (1, ) and foci at (º7, ) and (4, ) GRAPHING Graph the equation. Identify the important characteristics of the graph, such as the center, vertices, and foci. 1. ( º 6) + (y º ) = 4. ( + 7) = 1(y º 3) 3. (y º 8) º ( + 3) = 1 4. ( º 3) + (y + 6) = ( + 1) 5. + y = º (y + 4) 1 6 = 1 7. ( + 7) + (y º 1) = 1 8. (y º 4) = 3( + ) 68 Chapter 10 Quadratic Relations and Conic Sections

44 -, -, * * -, -, - * *,, * * " # $ % & ' ( ) CLASSIFYING Classify the conic section y + 36 º 4y + 36 = º 4y + 3 º 6y º 30 = º 9y + 18y + 3 = y º 10 º y + 10 = y º 5 + y + = y º y º 60 = y º y + 6 = y º 18 º 0y + 8 = º + 8y + 9 = y º 8y º = y º y º 37 = º y y + 55 = y º 4 º y º 4 = y + 4 º 36y + 3 = y º y + 63 = º y + 17 = 0 MATCHING Match the equation with its graph º 4y + 36 º 4y º 36 = y º y º = y y + 36 = y º + 6y = y º y + 61 = y º 4 + 6y + 4 = 0 A. B. C. / D. E. F FOCUS ON APPLICATIONS WHISPER DISHES are two parabolic dishes set up facing directly toward each other. A person listening at the focus of one dish is able to hear even the softest sound made at the focus of the other dish. REAL LIFE CLASSIFYING AND GRAPHING Classify the conic section and write its equation in standard form. Then graph the equation. 51. y º 1y = y º 6 º 8y + 4 = º y º 7 + 8y = y º 48 º 4y + 48 = y º º 8y + 1 = y º 1 º 4y + 36 = º y + 16y º 18 = y y + 7 = y º 1 º 1y + 36 = y º º 0y + 94 = º 8y + 1 = 0 6. º9 + 4y º 36 º 16y º 164 = WHISPER DISHES The whisper dish shown at the left can be seen at the Thronateeska Discovery Center in Albany, Georgia. Two dishes are positioned so that their vertices are 50 feet apart. The focus of each dish is 3 feet from its verte. Write equations for the cross sections of the dishes so that the verte of one dish is at the origin and the verte of the other dish is on the positive -ais Graphing and Classifying Conics 69

45 " # $ % & ' ( ) Test Preparation Challenge 64. FIGURE SKATING To practice making a figure eight, a figure skater will skate along two circles etched in the ice. Write equations for two eternally tangent circles that are each 6 feet in diameter so that the center of one circle is at the origin and the center of the other circle is on the positive y-ais. 65. VISUAL THINKING A new crayon has a cone-shaped tip. When it is used for the first time, a flat spot is worn on the tip. The edge of the flat spot is a conic section, as shown. What type(s) of conic could it be? 66. VISUAL THINKING When a pencil is sharpened the tip becomes a cone. On a pencil with flat sides, the intersection of the cone with each flat side is a conic section. What type of conic is it? 67. ASTRONOMY A Gregorian telescope contains two mirrors whose cross sections can be modeled by the equations y º 95,45 = 0 and º10y º 1440 = 0. What types of mirrors are they? 68. MULTIPLE CHOICE Which of the following is an equation of the hyperbola with vertices at (3, 5) and (3, º1) and foci at (3, 7) and (3, º3)? A ( º 3) º (y º ) = B (y º ) º ( º 3) = C (y º ) º ( º 3) = D ( º 3) º (y º ) = E (y º ) º ( º 3) = MULTIPLE CHOICE What conic does 5 + y º 100 º y + 76 = 0 represent? A Parabola B Circle C Ellipse D Hyperbola E Not enough information conic section conic section 70. DEGENERATE CONICS A degenerate conic occurs when the intersection of a plane with a double-napped cone is something other than a parabola, circle, ellipse, or hyperbola. EXTRA CHALLENGE a. Imagine a plane perpendicular to the ais of a doublenapped cone. As the plane passes through the cone, the intersection is a circle whose radius decreases and then increases. At what point is the intersection something other than a circle? What is the intersection? b. Imagine a plane parallel to the ais of a double-napped cone. As the plane passes through the cone, the intersection is a hyperbola whose vertices get closer together and then farther apart. At what point is the intersection something other than a hyperbola? What is the intersection? c. Imagine a plane parallel to the nappe passing through a double-napped cone. As the plane passes through the cone, the intersection is a parabola that gets narrower and then flips and gets wider. At what point is the intersection something other than a parabola? What is the intersection? nappe 630 Chapter 10 Quadratic Relations and Conic Sections

46 " # $ % & ' ( ) MIXED REVIEW SYSTEMS Solve the system using any algebraic method. (Review 3. for 10.7) 71. º y = y = y = 3 º y = 5 º3 º 6y = y = º 3y = = = y + 6y = y = º 18y = 17 EVALUATING LOGARITHMIC EXPRESSIONS Evaluate the epression. (Review 8.4) 77. log log log log 1/ log log SOLVING EQUATIONS Solve the equation. (Review 8.8) = = = e º e º 1 + 8e º e º6 = e º4 = e º3 = 7 History of Conic Sections INTERNET APPLICATION LINK THEN NOW IN 00 B.C. conic sections were studied thoroughly for the first time by a Greek mathematician named Apollonius. Si hundred years later, the Egyptian mathematician Hypatia simplified the works of Apollonius, making it accessible to many more people. For centuries, conics were studied and appreciated only for their mathematical beauty rather than for their occurrence in nature or practical use. TODAY astronomers know that the paths of celestial objects, such as planets and comets, are conic sections. For eample, a comet s path can be parabolic, hyperbolic, or elliptical. Tell what type of path each comet follows. Which comet(s) will pass by the sun more than once? , y º 13,050 = y º 13,00 º 18,400y + 1,900 = º 6500y + 0,000 º 5,000y º 695,000 = 0 Hypatia simplifies Apollonius Conics. A.D. 400 Debra Fischer discovers two planets B.C. Apollonius studies conic sections Johannes Kepler discovers that the planets orbits are elliptical Graphing and Classifying Conics 631

47 : ; 9 < Solving Quadratic Systems What you should learn GOAL 1 Solve systems of quadratic equations. GOAL Use quadratic systems to solve real-life problems, such as determining when one car will catch up to another in E. 58. Why you should learn it To model real-life situations with quadratic systems, such as finding the epicenter of an earthquake in Eample 4. REAL LIFE GOAL 1 SOLVING A SYSTEM OF EQUATIONS In Lesson 3. you studied two algebraic techniques for solving a system of linear equations. You can use the same techniques (substitution and linear combination) to solve quadratic systems. E X A M P L E 1 Finding Points of Intersection Find the points of intersection of the graphs of + y = 13 and y = + 1. To find the points of intersection, substitute + 1 for y in the equation of the circle. + y = 13 Equation of circle + ( + 1) = 13 Substitute + 1 for y = 13 + º 1 = 0 Epand the power. Combine like terms. ( º )( + 3) = 0 Factor. = or = º3 Zero product property You now know the -coordinates of the points of intersection. To find the y-coordinates, substitute = and = º3 into the linear equation and solve for y. The points of intersection are (, 3) and (º3, º). CHECK You can check your answer algebraically by substituting the coordinates of the points into each equation. Another way to check your answer is to graph the two equations. You can see from the graph shown that the line and the circle intersect in two points, at (, 3) and at (º3, º). ( 3, ) (, 3) ACTIVITY Developing Concepts Investigating Points of Intersection STUDENT HELP Look Back For help with solving systems, see p The circle and line in Eample 1 intersect in two points. A circle and a line can also intersect in one point or no points. Sketch eamples to illustrate the different numbers of points of intersection that the following graphs can have. a. Circle and parabola b. Ellipse and hyperbola c. Circle and ellipse d. Hyperbola and line 63 Chapter 10 Quadratic Relations and Conic Sections

48 : ; : E ; 9 < < = A B C D E X A M P L E Solving a System by Substitution Find the points of intersection of the graphs in the system. + 4y º 4 = 0 Equation 1 ºy + + = 0 Equation Because Equation has no -term, solve that equation for. ºy + + = 0 = y º Net, substitute y º for in Equation 1 and solve for y. + 4y º 4 = 0 Equation 1 (y º ) + 4y º 4 = 0 Substitute for. STUDENT HELP Look Back For help with factoring, see p y 4 º 8y y º 4 = 0 4y 4 º 4y = 0 4y (y º 1) = 0 4y (y º 1)(y + 1) = 0 y = 0, y = 1, or y = º1 Epand the power. Combine like terms. Factor common monomial. Difference of squares. Zero product property (, 0) (0, 1) (0, 1) The corresponding -values are = º, = 0, and = 0. The graphs intersect at (º, 0), (0, 1), and (0, º1), as shown. E X A M P L E 3 Solving a System by Linear Combination Find the points of intersection of the graphs in the system. + y º = 0 Equation 1 º y º 9 = 0 Equation You can eliminate the y -term by adding the two equations. The resulting equation can be solved for because it contains no other variables. + y º = 0 º y º 9 = 0 º = 0 Add. ( º 3)( º 5) = 0 Factor. = 3 or = 5 Zero product property The corresponding y-values are y = 0 and y = ±4. The graphs intersect at (3, 0), (5, 4), and (5, º4), as shown. (5, 4) (5, 4) (3, 0) 10.7 Solving Quadratic Systems 633

49 : ; : < = A B C D FOCUS ON CAREERS GOAL SOLVING QUADRATIC SYSTEMS IN REAL LIFE E X A M P L E 4 Solving a System of Quadratic Models SEISMOLOGIST A seismologist determines the location and intensity of an earthquake using an instrument which measures energy waves resulting from movements in the Earth s crust. REAL CAREER LINK INTERNET LIFE SEISMOLOGY A seismograph measures the intensity of an earthquake. Although a seismograph can determine the distance to the earthquake s epicenter, it cannot determine in what direction the epicenter is located. Use the following information from three seismographs to find an earthquake s epicenter. Location 1: 500 miles from the epicenter Location : 100 miles west and 400 miles south of Location miles from the epicenter Location 3: 300 miles east and 600 miles south of Location 1 00 miles from the epicenter Let each unit represent 100 miles. If Location 1 is at (0, 0), then Location is at (º1, º4) and Location 3 is at (3, º6). Write the equation of each circle. Location 1: + y = 5 Location : ( + 1) + (y + 4) = 16, or y + 8y + 16 = 16 Location 3: ( º 3) + (y + 6) = 4, or º y + 1y + 36 = 4 Subtract the equation for Location 1 from the equation for Location y + 8y + 16 = 16 º ( + y = 5) + 8y + 17 = º9 + 8y = º6, or + 4y = º13 Then subtract the equation for Location 1 from the equation for Location 3. º y + 1y + 36 = 4 º ( + y = 5) º6 + 1y + 45 = º1 º6 + 1y = º66, or º + y = º11 You are left with two linear equations. Solve this linear system to find the epicenter. + 4y = º13 º + y = º11 6y = º4 y = º4 = 3 The epicenter of the earthquake is 300 miles east and 400 miles south of Location Chapter 10 Quadratic Relations and Conic Sections

50 = A B C D GUIDED PRACTICE Vocabulary Check Concept Check 1. Complete this statement: The equations + 3y º y = 4 and + y = 5 are an eample of a(n)? system.. Sketch an eample of a circle and a line intersecting in a single point. 3. Eplain what method you would use to find the points of intersection of the graphs in the following system. Do not solve the system. 4 + y º 16 = 0 Equation 1 Skill Check º y + 7 = 0 Equation Find the points of intersection, if any, of the graphs in the system y = y + 8 º 0y + 7 = 0 y = y y + 7 = y º 3 = 8 7. º + y + = 0 º y = 10 º + º y + 3 = 0 8. SEISMOLOGY Look back at Eample 4. Why are three (not just two) seismographs needed to determine the location of the epicenter? PRACTICE AND APPLICATIONS STUDENT HELP Etra Practice to help you master skills is on p STUDENT HELP HOMEWORK HELP Eample 1: Es. 9 3 Eamples, 3: Es Eample 4: Es. 5 55, CHECKING POINTS OF INTERSECTION Determine whether the given point is a point of intersection of the graphs in the system y = y = º 4y º 16 = 0 y = º3 y = º º 1 º + y + 1 = 0 Point: (º3, 4) Point: (4, º5) Point: (6, 11) 1. 3 º 5y + y = º 4y = º 5 + 8y + y = 3 y = + 10 y = º + 3 y = º 1 Point: (º3, 4) Point: (º5, 7) Point: (, 1) SOLVING SYSTEMS Find the points of intersection, if any, of the graphs in the system. 15. º y = y = º3 + y = 9 º3 + y = º7 º y = 0 º + y = y = y = y = º6 º + y = º4 y = º + 8y = y = º 5y = y = 15 º + y = º1 3 + y = 6 + y = y = y = y = 5 + y = º1 y = º 4 y = = 6y 8. + y = y = 7 y = º º 3y = 3 y = º y º = y = º y = º6 y = º y = º + y = Solving Quadratic Systems 635

51 F G H I J K L M POLICE OFFICER The duties of a police officer vary. An officer in a large city is often assigned to a specific type of duty, while an officer in a small community usually performs a variety of tasks. REAL CAREER LINK INTERNET LIFE FOCUS ON CAREERS SOLVING SYSTEMS Find the points of intersection, if any, of the graphs in the system y = º3 + y º 3 = º + y + 10 = 0 º 5y = 5 º y + 7 = 0 º3y = y º 10 = y = y = 4y = 0 4 º y = º4 º 6 = ºy 39. y + = 40. º 16y = y = y = 8 + y = 9 + y = º y + 16y º 18 = º y º 8 + 8y º 4 = 0 y º 48 º 16y º 3 = 0 + y º 8 º 8y + 4 = y º 4 º 8y + 4 = º y = 0 + 4y º 4 = y º 64 = y º = º 4y º 0 º 64y º 17 = 0 º y º 9 = y º y = º y º 10 = º y º 8 + 6y º 9 = 0 y º + 3 = 0 º 3y y º 43 = º 5y º 100 = º º y º 4 = 0 y º + 16 = 0 + 3y º 4y º 10 = 0 SYSTEMS OF THREE EQUATIONS Find the points, if any, that the graphs of all three equations have in common y = y º 8 = 0 + y y º 5 = 0 + y º 3 + y = 0 + y = 1 + y º 5 º 10 = y = y º 4 º 4y = y = 16 + y º 4 = 54 y = º y = 3 º CRITICAL THINKING Suppose a line intersects a circle whose center is at the origin, and the line passes through the origin. If you know one of the points of intersection, how do you know what the other point of intersection is without solving the system algebraically? 57. LOGICAL REASONING Sketch eamples to illustrate the different numbers of points of intersection that a circle and an ellipse can have if both are centered at the origin. 58. LAW ENFORCEMENT Suppose a car is traveling down the highway at a constant rate of 60 miles per hour. It passes a police car parked at the side of the road. To catch up to the car, the police officer accelerates at a constant rate. The distance d (in miles) the police car has traveled as a function of time t (in hours) since the other car has passed it is given by d = 3600t. Write and solve a system of equations to calculate how long it takes the police car to catch up to the other car. 59. COMMUNICATIONS The range of a radio station is bounded by a circle given by the following equation: + y º 160 = 0 A straight highway can be modeled by the following equation: y = º Find the length of the highway that lies within the range of the radio station. 636 Chapter 10 Quadratic Relations and Conic Sections

52 O P N Q F G H I J K L M 60. BUS BOUNDARY To be eligible to ride the school bus to East High School, a student must live at least 1 mile from the school. How long is the portion of Clark Street for which the residents are not eligible to ride the school bus? (Use a coordinate plane in which the school is at (0, 0) and each unit represents one mile.) mi 5 mi Main St. STUDENT HELP HOMEWORK HELP Visit our Web site for help with problem solving in Es INTERNET 1 mi State St. East High School Clark St. 61. NAVIGATION LORAN (Long-Distance Radio Navigation) uses synchronized pulses sent out by pairs of transmitting stations. By calculating the difference in the times of arrival of the pulses from two stations, the LORAN equipment on a ship locates the ship on a hyperbola. By doing the same thing with a second pair of stations, LORAN locates the ship at the intersection of two hyperbolas. Suppose LORAN equipment indicates that a ship s location is the point of intersection of the graphs in the following system: y º 4 = 0 º 5y = 0 Find the ship s location given that it is north and east of the origin. 6. HYPERBOLIC MIRROR In a hyperbolic mirror, light rays directed to one focus will be reflected to the other focus. The mirror shown at the right has the following equation: y º = At which point on the mirror will light from the point (0, 8) be reflected to the focus at (º10, 0)? (0, 8) ( 10, 0) (10, 0) Test Preparation Challenge 63. EARTHQUAKES An earthquake occurred in Peru on April 18, Use the following information to approimate the location of the epicenter. Source: U.S. Department of the Interior Geological Survey Location 1: (Cayambe, Ecuador) The epicenter was 1300 kilometers away. Location : (Cocohabamba, Bolivia, 100 kilometers east and 1900 kilometers south of Cayambe) The epicenter was 1300 kilometers away. Location 3: (Cerro El Oso, Venezuela, 1100 kilometers east and 1000 kilometers north of Cayambe) The epicenter was 500 kilometers away. 64. MULTIPLE CHOICE How many points of intersection do the equations + y = 6 and + 4y = 7 have? A 0 B 1 C D 3 E MULTIPLE CHOICE Which of the following is a point of intersection of the graphs of y º 900 = 0 and º + y + 5 = 0? A (º5, 0) B (0, 5) C (, 5) D (1, 5) E (0, º5) 66. CRITICAL THINKING Write equations for three different conics that all intersect at the point (º4, 6) Solving Quadratic Systems 637

53 R S T U V W X Y MIXED REVIEW EVALUATING EXPRESSIONS Evaluate the epression for the given value of. (Review 1. for 11.1) when = º 1 when = (º) º 1 when = when = 4 (º3 ) º WRITING FUNCTIONS Write a polynomial function of least degree that has real coefficients, the given zeros, and a leading coefficient of 1. (Review 6.7) 71. 3, º3, ,,, i, ºi i, 3 º i 75., º1, º1 º i 76. º, º3, i, i GRAPHING Graph the function. Then state the domain and range. (Review 7.5) 77. ƒ() = ƒ() = 5 º ƒ() = º( + 4) 1/ ƒ() = º ƒ() = ƒ() = 5( º 1) 1/3 CLASSIFYING CONICS Classify the conic section. (Review 10.6) y + + y = º y º 8 + 4y º 9 = º y + 13 = y º + 6y + 9 = 0 QUIZ 3 Self-Test for Lessons 10.6 and 10.7 Write an equation for the conic section. (Lesson 10.6) 1. Circle with center at (º3, º5) and radius 8. Ellipse with vertices at (º7, ) and (6, ) and foci at (4, ) and (º5, ) 3. Parabola with verte at (4, º1) and focus at (7, º1) 4. Hyperbola with foci at (, º1) and (, 8) and vertices at (, 3) and (, 4) Classify the conic section. (Lesson 10.6) y º 8 + 3y + 1 = 0 6. º3 º 3y y + 1 = 0 7. ºy + + 5y + 6 = 0 8. º6 + 4y = 0 Find the points of intersection, if any, of the graphs in the system. (Lesson 10.7) 9. 3 º 4 º y + = º + y + 4 º 6y + 4 = 0 y = º y º 4 º 6y + 1 = y + 4y º 1 = 0 1. y º 6 º y º 3 = 0 º 16y º 64y º 80 = 0 y º 4y = SEISMOLOGY A seismograph records the epicenter of an earthquake 50 miles away. A second seismograph, 50 miles west and 35 miles north of the first, records the epicenter as being 35 miles away. A third seismograph, 80 miles due west of the first, records the epicenter 30 miles away. Where was the earthquake s epicenter in relation to the first seismograph? (Lesson 10.7) 638 Chapter 10 Quadratic Relations and Conic Sections

54 Z [ \ ] ^ _ ` a C H A P T E R 1 0 Etension What you should learn GOAL Find the eccentricity of a conic section. Why you should learn it To write equations for real-life conics, such as the moon s orbit in Eample 3. Eccentricity of Conic Sections Some ellipses are more oval than others. In an ellipse that is nearly circular, the ratio c:a is close to 0. In a more oval ellipse, c:a is close to 1. This ratio is called the eccentricity of the ellipse. Every conic has an eccentricity e associated with it. CONCEPT SUMMARY ECCENTRICITY OF CONIC SECTIONS Let c be the distance from each focus to the center of the conic section, and let a be the distance from each verte to the center. c a major ais minor ais c a The eccentricity of an ellipse is e = a c, and 0 < e < 1. The eccentricity of a hyperbola is e = a c, and e > 1. The eccentricity of a parabola is e = 1. The eccentricity of a circle is e = 0. E X A M P L E 1 Finding Eccentricity Earth, as seen from the moon Find the eccentricity of the conic section described by the equation. a. ( + ) = 4(y º 1) b. 5( + ) º 36(y º 1) = 900 a. This equation describes a parabola. By definition, the eccentricity is e = 1. b. This equation describes a hyperbola with a = 3 6 = 6, b = 5 = 5, and c = a + b = 6 1. The eccentricity is e = a c = E X A M P L E Using Eccentricity to Write an Equation Find an equation of the hyperbola with center (3, º5), verte (9, º5), and e =. Use the form ( º h) (y º k) a º b = 1. The verte lies 9 º 3 = 6 units from the c c center, so a = 6. Because e = =, you know that 6 =, or c = 1. Therefore, a b = c º a = 144 º 36 = 108. The equation is ( º 3) º ( y + 5) = Chapter 10 Etension 639