TEKS 9.4 a.5, A.5.B, A.5.C Before Now Graph and Write Equations of Ellipses You graphed and wrote equations of parabolas and circles. You will graph and write equations of ellipses. Wh? So ou can model an elliptical region, as in Eample 3. Ke Vocabular ellipse foci vertices major ais center co-vertices minor ais An ellipse is the set of all points P in a plane such that the sum of the distances between P and two fied points, called the foci, is a constant. The line through the foci intersects the ellipse at the two vertices. The major ais joins the d d 5 constant vertices. Its midpoint is the ellipse s center. The line perpendicular to the major ais at the center intersects the ellipse at the two co-vertices, which are joined b the minor ais. In this chapter, ellipses have a horizontal or a vertical major ais. d d P IDENTIFY AXES Observe that the major ais of an ellipse contains the foci and is alwas longer than the minor ais. verte (a, 0) major ais (c, 0) (0, b) (c, 0) (0, b) verte (a, 0) minor ais (b, 0) major ais verte (0, a) (0, c) (0, c) (b, 0) verte (0, a) minor ais Ellipse with horizontal major ais a b 5 Ellipse with vertical major ais b a 5 KEY CONCEPT For Your Notebook Standard Equation of an Ellipse with Center at the Origin Equation Major Ais Vertices Co-Vertices 5 Horizontal (6a, 0) (0, 6b) a b 5 Vertical (0, 6a) (6b, 0) b a The major and minor aes are of lengths a and b, respectivel, where a > b > 0. The foci of the ellipse lie on the major ais at a distance of c units from the center, where c 5 a b. 634 Chapter 9 Quadratic Relations and Conic Sections
E XAMPLE Graph an equation of an ellipse Graph the equation 4 5 5 00. Identif the vertices, co-vertices, and foci of the ellipse. Solution STEP Rewrite the equation in standard form. 4 5 5 00 Write original equation. 4 00 5 00 5 00 00 Divide each side b 00. ANOTHER WAY You can graph the ellipse using a graphing calculator b solving for to obtain 56Î 5 and then entering this equation as two separate functions. STEP STEP 3 5 4 5 Simplif. Identif the vertices, co-vertices, and foci. Note that a 5 5 and b 5 4, so a 5 5 and b 5. The denominator of the -term is greater than that of the -term, so the major ais is horizontal. The vertices of the ellipse are at (6a, 0) 5 (65, 0). The co-vertices are at (0, 6b) 5 (0, 6). Find the foci. c 5 a b 5 5 5, so c 5 Ï The foci are at (6Ï, 0), or about (64.6, 0). Draw the ellipse that passes through each verte and. at classzone.com (, 0) 3 (0, ) (, 0) (5, 0) (0, ) (5, 0) GUIDED PRACTICE for Eample Graph the equation. Identif the vertices, co-vertices, and foci of the ellipse.. 6 9 5. 36 49 5 3. 5 9 5 5 E XAMPLE Write an equation given a verte and a Write an equation of the ellipse that has a verte at (0, 4), a at (3, 0), and center at (0, 0). Solution Sketch the ellipse as a check for our final equation. B smmetr, the ellipse must also have a verte at (0, 4) and a at (3, 0). Because the verte is on the -ais and the is on the -ais, the major ais is vertical with a 5 4, and the minor ais is horizontal with b 5 3. c An equation is 5, or 5. 3 4 9 6 (3, 0) verte (0, 4) verte (0, 4) (3, 0) 9.4 Graph and Write Equations of Ellipses 635
E XAMPLE 3 TAKS REASONING: Multi-Step Problem LIGHTNING When lightning strikes, an elliptical region where the strike most likel hit can often be identified. Suppose it is determined that there is a 50% chance that a lightning strike hit within the elliptical region shown in the diagram. Write an equation of the ellipse. The area A of an ellipse is A 5 πab. Find the area of the elliptical region. 400 m 00 m ANOTHER WAY For an alternative method for solving the problem in Eample 3, turn to page 640 for the Problem Solving Workshop. Solution STEP The major ais is horizontal, with a 5 400 5 00 and b 5 00 5 00. STEP An equation is 5, or 5. 00 00 40,000 0,000 The area is A 5 π(00)(00) ø 6,800 square meters. E XAMPLE 4 Write an equation given a verte and a Write an equation of the ellipse that has a verte at (8, 0), a at (4, 0), and center at (0, 0). Solution Make a sketch of the ellipse. Because the given verte and lie on the -ais, the major ais is horizontal, with a 5 8 and c 5 4. To find b, use the equation c 5 a b. 4 5 8 b verte (8, 0) (4, 0) 8 (4, 0) verte (8, 0) b 5 8 4 5 48 b 5 Ï 48, or 4Ï 3 c An equation is 8 5, or 5. (4Ï 3 ) 64 48 GUIDED PRACTICE for Eamples, 3, and 4 Write an equation of the ellipse with the given characteristics and center at (0, 0). 4. Verte: (7, 0); : (0, ) 5. Verte: (0, 6); : (5, 0) 6. Verte: (0, 8); : (0, 3) 7. Verte: (5, 0); : (3, 0) 8. WHAT IF? In Eample 3, suppose that the elliptical region is 50 meters from east to west and 350 meters from north to south. Write an equation of the elliptical boundar and find the area of the region. 636 Chapter 9 Quadratic Relations and Conic Sections
9.4 EXERCISES SKILL PRACTICE HOMEWORK KEY 5 WORKED-OUT SOLUTIONS on p. WS for Es., 9, and 49 5 TAKS PRACTICE AND REASONING Es. 35, 45, 46, 5, 5, 54, and 55. VOCABULARY Cop and complete: An ellipse is the set of all points P such that the sum of the distances between P and two fied points, called the?, is a constant.. WRITING Describe how to find the foci of an ellipse given the coordinates of its vertices and co-vertices. EXAMPLE on p. 635 for Es. 3 6 GRAPHING Graph the equation. Identif the vertices, co-vertices, and foci of the ellipse. 3. 6 4 5 4. 4 5 5 5. 9 49 5 6. 44 64 5 7. 400 8 5 8. 36 5 5 9. 4 5 36 0. 9 5 9. 6 9 5 44. 5 49 5 5 3. 6 5 5 600 4. 7 8 5 648 ERROR ANALYSIS Describe and correct the error in graphing the ellipse. 5. 4 6 5 6. 3 5 EXAMPLES and 4 on pp. 635 636 for Es. 7 35 WRITING EQUATIONS Write an equation of the ellipse with the given characteristics and center at (0, 0). 7. Verte: (5, 0) 8. Verte: (0, 0) 9. Verte: (4, 0) Co-verte: (0, 3) Co-verte: (6, 0) Co-verte: (0, 9) 0. Verte: (0, 6). Verte: (0, ). Verte: (0, 0) Co-verte: (4, 0) Co-verte: (, 0) Co-verte: (0, 6) 3. Verte: (0, 8) 4. Verte: (4, 0) 5. Verte: (0, 9) Focus: (0, 6) Focus: ( Ï 7, 0) Focus: (0, 4Ï ) 6. Verte: (5, 0) 7. Verte: (0, 4) 8. Verte: (3, 0) Focus: (3, 0) Focus: (0, Ï 3 ) Focus: (4Ï 3, 0) 9. Co-verte: (0, Ï 7 ) 30. Co-verte: (3Ï 5, 0) 3. Co-verte: (0, 5Ï 7 ) Focus: (3, 0) Focus: (0, 6) Focus: (5, 0) 3. Co-verte: (0, 5) 33. Co-verte: ( Ï 5, 0) 34. Co-verte: (3, 0) Focus: (8, 0) Focus: (0, 4) Focus: (0, 4) 9.4 Graph and Write Equations of Ellipses 637
35. TAKS REASONING What is an equation of the ellipse with center at the origin, a verte at (0, ), and a at (8, 0)? A 5 B 5 C 5 D 5 44 64 64 44 8 8 GRAPHING In Eercises 36 44, the equations of parabolas, circles, and ellipses are given. Graph the equation. 36. 5 64 37. 5 8 5 05 38. 36 5 0 39. 65 5 30 40. 30 30 5 480 4. 4. 3 48 4 400 5 43. 4 5 0 75 5 64 64 5 4 44. 6 0 5 60 45. TAKS REASONING Consider the graph of 5. Describe the effects 9 5 on the graph of changing the denominator of the -term first from 5 to 9 and then from 9 to 4. Graph the original equation and the two revised equations in the same coordinate plane. 46. TAKS REASONING Write an equation of an ellipse in standard form. Graph the equation on a graphing calculator b rewriting it as two functions. Give a viewing window that does not distort the shape of the ellipse, and eplain how ou found our viewing window. 47. CHALLENGE Use the definition of an ellipse to show that c 5 a b for an ellipse with equation 5 and foci at (c, 0) and (c, 0). (Hint: Draw a a b diagram. Consider the point P(a, 0) on the ellipse.) PROBLEM SOLVING EXAMPLE 3 on p. 636 for Es. 48 50 48. MARS On Januar 3, 004, the Mars rover Spirit bounced on its airbags to a landing within Gusev crater. Scientists had estimated that there was a 99% chance the rover would land inside an ellipse with a major ais 8 kilometers long and a minor ais kilometers long. Write an equation of the ellipse. Then find its area. Artist s rendering of landing 49. AUSTRALIAN FOOTBALL The plaing field for Australian football is an ellipse that is between 35 and 85 meters long and between 0 and 55 meters wide. Write equations of ellipses with vertical major aes that model the largest and smallest fields described. Then write an inequalit that describes the possible areas of these fields. 638 5 WORKED-OUT SOLUTIONS on p. WS 5 TAKS PRACTICE AND REASONING
50. HEALTH CARE A lithotripter uses shock waves to break apart kidne stones or gallstones inside the bod. Shock waves generated at one of an ellipsoid (a three-dimensional shape with an elliptical cross section) reflect to the stone positioned at the second. Write an equation for the cross section of the ellipsoid with the dimensions shown. How far apart are the foci? 5. TAKS REASONING Halle s comet ranges from 0.59 to 35.3 astronomical units from the sun, which is at one of the comet s elliptical orbit. (An astronomical unit is Earth s mean distance from the sun.) Eplain using a sketch how to find a and c. Then write an equation for the orbit. 5. TAKS REASONING A small airplane with enough fuel to fl 600 miles safel will take off from airport A and land at airport B, 450 miles awa. a. Reason The region in which the airplane can fl is bounded b an ellipse. Eplain wh this is so. b. Calculate Let (0, 0) represent the center of the ellipse. Find the coordinates of each airport. c. Appl Suppose the plane flies from airport A straight past airport B to a verte of the ellipse and then straight back to airport B. How far does the plane fl? Use our answer to find the coordinates of the verte. d. Model Write an equation of the ellipse. 53. CHALLENGE An art museum worker leaves an 8-foot-tall painting leaning against a wall. Later, the top of the painting slides down the wall, and the painting falls to the floor. Use the diagram to find an equation of the path of the point (, ) as the painting falls. (0, ) ft (, ) (, 0) airport A 6 ft airport B MIXED REVIEW FOR TAKS TAKS PRACTICE at classzone.com REVIEW TAKS Preparation p. 66; TAKS Workbook 54. TAKS PRACTICE Iris wants to make candles shaped like rectangular prisms that measure 3 inches long, inches wide, and 5 inches high. To make the candles, she melts the clindrical block of wa shown. How man candles can she make? TAKS Obj. 0 A 3 B 6 C 36 D 88 8.5 in. 8 in. REVIEW Lesson 5.; TAKS Workbook 55. TAKS PRACTICE The area of a triangle is 45m 7 n 3 square units and its height is 5m 0 n 9 units. What is the length of the triangle s base? TAKS Obj. 5 F 3m3 n 4 units G 6n4 m 3 units H 3m 3 n 4 units J 6m 7 n units EXTRA PRACTICE for Lesson 9.4, p. 08 ONLINE QUIZ at classzone.com 639
LESSON 9.4 TEKS a.4, a.5, a.6, A.5.C Using ALTERNATIVE METHODS Another Wa to Solve Eample 3, page 636 MULTIPLE REPRESENTATIONS In the second part of Eample 3 on page 636, ou found the area of an ellipse using a formula. You can also approimate the area of an ellipse b summing the areas of rectangles. P ROBLEM LIGHTNING When lightning strikes, an elliptical region where the strike most likel hit can often be identified. Suppose it is determined that there is a 50% chance that a lightning strike hit within the elliptical region shown in the diagram. Write an equation of the ellipse. Find the area of the elliptical region. M ETHOD Summing Rectangles As ou saw on page 636, the ellipse has the equation 5. Approimate the area of the ellipse as follows. 00 00 STEP Graph the first-quadrant portion of the ellipse. Then draw rectangles of width 40 and height equal to the -value of the ellipse at the rectangle s left edge. The first rectangle s height is 5 00. To find the other -values, solve for to obtain 5 Ï 00. Use a calculator to get 4 ø 98.0, 3 ø 9.7, 4 5 80, and 5 5 60. 40 40 00 40 40 40 3 4 5 0 0 40 80 0 60 00 STEP Calculate the total area A of the rectangles. A ø 40(00) 40(98.0) 40(9.7) 40(80) 40(60) 5 7,88 m STEP 3 Multipl the total area of the rectangles b 4 to obtain an estimate of 4(7,88) ø 68,800 square meters for the area of the ellipse. P RACTICE. Solve the problem above using rectangles of width 0. Is this estimate better or worse than the estimate above? Eplain.. REASONING Eplain using our results from Eercise how to obtain a closer and closer approimation of the ellipse s area. 3. WHAT IF? Suppose that the ellipse in the problem had a horizontal major ais of 50 meters and a minor ais of 00 meters. a. Write an equation of the ellipse. b. Use the method above to approimate the area of the ellipse. 640 Chapter 9 Quadratic Relations and Conic Sections
MIXED REVIEW FOR TEKS TAKS PRACTICE Lessons 9. 9.4 MULTIPLE CHOICE. PARABOLIC REFLECTORS Parabolic reflectors with a microphone at the allow the operator to listen to sounds from far awa. A certain parabolic microphone has a reflector that is.4 inches in diameter and 6 inches deep. Approimatel how far is the from the verte? TEKS A.5.B A 5. inches B 6 inches C. inches D 4.8 inches. RADAR An anchored fishing boat s radar has a range of 6 miles. A second boat 6 miles north and 4 miles east of the fishing boat begins moving westward. For approimatel what distance will the second boat be in radar range of the first boat? TEKS A.5.B F 4.8 miles G 8.8 miles H.5 miles J 9.6 miles 3. PLANETARY ORBIT In its elliptical orbit, Mercur ranges from 9 million miles to 44 million miles from the sun. The sun is at one of the orbit. Which equation could represent Mercur s orbit? TEKS A.5.B 5. ACCIDENT INVESTIGATION A car skids while turning to avoid an accident. The circular skid mark is shown below. The car s speed v (in meters per second) can be approimated b v 5 Ï 9.8mr where r is the radius (in meters) of the skid mark and m is a constant that depends on the road surface and weather conditions (0 m ). About how fast was the car traveling if it is determined that m 5 0.7? TEKS a.4 v 0 6 (0, 0) 4 6 8 0 4 6 r A 9.4 m/s (0, ) B 9.8 m/s C 0.0 m/s D 8.3 m/s 4 (6, ) 6. TANGENT LINES Two lines are tangent to the circle 5 3, one at (, 3) and one at (3, ). What is the relationship between the two lines? TEKS a.4 F The two lines are parallel. G H classzone.com The two lines are perpendicular. The two lines intersect at the origin. A 5 (36.5) (7.5) J The two lines intersect at 0, 3. B C D 5 (44) (9) 5 (36.5) (9) 5 (36.5) (35.7) 4. DRIVING DISTANCE To get from our home to the beach, ou drive 8 miles south, then 6 miles east, and then 4 miles south. What is the straight-line distance from our home to the beach? TEKS A.5.B F 6.5 miles G 0 miles H.5 miles J 8 miles GRIDDED ANSWER 0 3 4 5 6 7 8 9 7. SOLAR COOKING You can make a solar hot dog cooker b shaping foil-lined cardboard into a parabolic trough and passing a wire through the of each end piece. For the trough shown, how far from the bottom, to the nearest tenth of an inch, should ou place the wire? TEKS A.5.B Mied Review for TEKS 64