Skills Practice Skills Practice for Lesson.1 Name Date Try to Stay Focused Ellipses Centered at the Origin Vocabulary Match each definition to its corresponding term. 1. an equation of the form a. ellipse x 2 2 a b 1 or x2 2 2 b a 1 2 2. the two fixed points of an ellipse b. foci / focus 3. the set of all points in a plane such c. general form that the sum of the distances from of an ellipse two fixed points is constant 4. the endpoints of the major axis d. standard form of an ellipse 5. an equation of the form e. major axis Ax 2 By 2 E 0 6. The segment that connects two f. vertices points on the ellipse, passes through the center of the ellipse, and is perpendicular to the major axis 7. the endpoints of the minor axis g. semi-major axis 8. the longest segment that connects h. minor axis two points on the ellipse and passes through the center of the ellipse 9. a segment that connects the center i. co-vertices of an ellipse to one of the vertices 10. a measure of the elongation of an j. semi-minor axis ellipse 11. a segment that connects the center k. eccentricity of an ellipse to one of its co-vertices Chapter l Skills Practice 811
Problem Set Given the equation of each ellipse in standard form, write the equation of the ellipse in general form. 1. x 2 64 36 1 2. x 2 25 576 ( x2 64 36 ) 1 9x 2 16y 2 576 9x 2 16y 2 576 0 9 1 3. x 2 81 9 1 4. x 2 25 64 1 5. x2 4 9 1 6. x2 9 16 1 Given the equation of each ellipse in general form, write the equation of the ellipse in standard form. 7. 16x 2 25y 2 400 0 8. 36x 2 9y 2 324 0 16x 2 25y 2 400 0 16x 2 25y 2 400 16x 2 400 25 400 400 400 x 2 25 16 1 8 Chapter l Skills Practice
Name Date 9. 81x 2 16y 2 96 0 10. 4x 2 9y 2 36 0 11. 25x 2 2y 2 50 0. 4x 2 16y 2 64 0 Given the x-intercepts and y-intercepts of an ellipse, sketch the ellipse. Label the major axis, vertices, semi-major axis, minor axis, co-vertices, and semi-minor axis. 13. x-intercepts: ( 4, 0), (4, 0) 14. x-intercepts: ( 3, 0), (3, 0) y-intercepts: (0, 5), (0, 5) y-intercepts: (0, 2), (0, 2) 8 y 6 vertex 4 major axis 2 semi-minor axis co-vertex co-vertex 8 6 4 2 2 4 6 8 x 2 semi-major 4 axis minor axis 6 vertex 8 Chapter l Skills Practice 813
15. x-intercepts: ( 5, 0), (5, 0) 16. x-intercepts: ( 8, 0), (8, 0) y-intercepts: (0, 3), (0, 3) y-intercepts: (0, 3), (0, 3) 17. x-intercepts: ( 3, 0), (3, 0) 18. x-intercepts: ( 5, 0), (5, 0) y-intercepts: (0, 8), (0, 8) y-intercepts: (0, 4), (0, 4) 814 Chapter l Skills Practice
Name Date Given the equation of an ellipse in standard form, sketch the ellipse. Label the vertices and co-vertices. Give the coordinates of the foci for each ellipse. 19. x 2 64 9 1 20. x 2 25 16 1 8 y 8 6 4 (0, 3) 2 ( 8, 0) (8, 0) 6 4 2 2 4 6 8 2 4 (0, 3) 6 8 x The major axis is along the x-axis. a 8 and b 3 c 2 a 2 b 2 c 2 64 9 c 55 7.42 The coordinates of the foci are (7.42, 0), ( 7.42, 0). Chapter l Skills Practice 815
21. x2 9 25 1 22. x 2 64 81 1 816 Chapter l Skills Practice
Name Date 23. x2 4 9 1 24. x2 9 36 1 Chapter l Skills Practice 817
Given the equation of an ellipse in standard form, calculate the eccentricity of each ellipse. Then sketch the ellipse. 25. x 2 25 36 1 26. x 2 16 9 1 8 y 6 4 2 8 6 4 2 2 4 6 8 2 x 4 6 8 The major axis is along the y-axis. a 6 and b 5 c 2 a 2 b 2 c 2 36 25 c 11 3.32 e c a 3.32 0.553 6 818 Chapter l Skills Practice
Name Date 27. x 2 36 100 1 28. x 2 81 9 1 Chapter l Skills Practice 819
29. x 2 49 9 1 30. x 2 10 20 1 820 Chapter l Skills Practice
Name Date Given the equation of an ellipse in standard form, determine the coordinates of the vertices, the coordinates of the co-vertices, the coordinates of the foci, and the eccentricity. Then graph the ellipse. 31. x 2 144 25 1 32. x 2 16 49 1 y 9 6 3 9 6 3 3 6 9 3 x 6 9 Vertices: (, 0), (, 0) Co-vertices: (0, 5), (0, 5) Foci: (10.91, 0), ( 10.91, 0) Eccentricity: 0.91 Chapter l Skills Practice 821
33. x 2 25 100 1 34. x 2 81 49 1 822 Chapter l Skills Practice
Name Date 35. x 2 49 64 1 36. x 2 100 36 1 Chapter l Skills Practice 823
824 Chapter l Skills Practice
Skills Practice Skills Practice for Lesson.2 Name Date It s Just a Jump to the Left! Ellipses Not Centered at the Origin Problem Set Write an equation to represent each circle. 1. a circle with its center at the origin and a radius of 4 units x 2 y 2 16 2. a circle with its center at (2, 1) and a radius of 4 units 3. a circle with its center at ( 4, 2) and a radius of 6 units 4. a circle with its center at ( 3, 5) and a radius of 5 units 5. a circle with its center at (0, 2) and a radius of 8 units 6. a circle with its center at ( 8, 1) and a radius of 3 units Chapter l Skills Practice 825
Identify the center of each ellipse. Then graph the ellipse. 7. (x 1)2 49 36 1 8. ( x 2)2 25 ( y 1)2 1 16 8 y 6 4 2 (1, 0) 8 6 4 2 2 4 6 8 2 x 4 6 8 (1, 0) 9. (x 4)2 ( y 3)2 1 10. 81 49 ( x 2)2 ( y 2)2 1 49 25 826 Chapter l Skills Practice
Name Date 11. (x 3)2 ( y 1)2 1. 4 9 ( x 4)2 ( y 1)2 1 9 16 Given the equation of an ellipse, determine the coordinates of the center, the coordinates of the vertices, the coordinates of the co-vertices, the coordinates of the foci, and the eccentricity. Then graph and label the ellipse. 13. (x 1)2 ( y 2)2 1 14. 144 49 8 y ( x 2)2 ( y 4)2 1 16 64 6 (1, 5) 4 2 9 6 3 3 6 9 x (10.75, 2) 2 (11, 2) ( 8.75, 2) (1, 2) (13, 2) 4 6 8 (1, 9) Center: (1, 2) Vertices: (13, 2), ( 11, 2) Co-vertices: (1, 5), (1, 9) Foci: (10.75, 2), ( 8.75, 2) Eccentricity: 0.81 Chapter l Skills Practice 827
15. (x 1)2 ( y 3)2 1 16. 64 100 ( x 2)2 ( y 5)2 1 81 25 828 Chapter l Skills Practice
Name Date 17. (x 4)2 ( y 3)2 1 18. 100 144 ( x 3)2 ( y 1)2 1 25 36 Chapter l Skills Practice 829
830 Chapter l Skills Practice
Skills Practice Skills Practice for Lesson.3 Name Date Graphs, Equations, and Key Characteristics of Ellipses Forms of Ellipses Problem Set Write an equation in standard form for each ellipse. Then graph the ellipse. 1. An ellipse with center at (0, 0), one vertex at ( 8, 0), and one co-vertex at (0, 3). 8 y 6 4 2 8 6 4 2 2 4 6 8 2 x 4 6 8 The distance between the center and a vertex is 8 units, so a 8. The distance between the center and a co-vertex is 3 units, so b 3. The vertex is along the x-axis. The standard form of an ellipse centered at the origin with the major axis along the x-axis is x2 a 1. 2 2 b So, the equation is x2 1. 64 9 Chapter l Skills Practice 831
2. An ellipse with vertices at ( 3, 2), (9, 2) and the length of the minor axis is 8 units. 832 Chapter l Skills Practice
Name Date 3. An ellipse with one vertex at ( 8, 3) and the foci at (10, 3) and ( 6, 3). Chapter l Skills Practice 833
4. An ellipse with foci at (6, 5) and (6, 7) and the length of the major axis is 20 units. 834 Chapter l Skills Practice
Name Date 5. An ellipse with center at (4, 2), one vertex at (4, 2), and one co-vertex at (7, 2). Chapter l Skills Practice 835
6. An ellipse with vertices at (1, 3) and ( 9, 3) and one focus at (0, 3). 836 Chapter l Skills Practice
Name Date 7. An ellipse with foci at ( 1, 4) and ( 1, 2) and the length of the minor axis is 8 units. Chapter l Skills Practice 837
8. An ellipse centered at ( 5, 3) with a vertical major axis and eccentricity of 8 10. 838 Chapter l Skills Practice
Name Date Write an equation in standard form for each ellipse. Then determine the coordinates of the center, the coordinates of the vertices, the coordinates of the co-vertices, the coordinates of the foci, and the eccentricity. Finally, graph the ellipse. 9. 16x 2 25y 2 400 0 10. 9x 2 16y 2 36x 32y 92 0 8 y 6 4 2 8 6 4 2 2 4 6 8 2 x 4 6 8 16x 2 25y 2 400 0 16x 2 25y 2 400 16 x 2 400 25 400 400 400 x 2 25 16 1 Center: (0, 0) Vertices: (5, 0), ( 5, 0) Co-vertices: (0, 4), (0, 4) Foci: (3, 0), ( 3, 0) Eccentricity: 0.6 Chapter l Skills Practice 839
11. 25x 2 16y 2 150x 32y 159 0. 25x 2 49y 2 200x 196y 629 0 840 Chapter l Skills Practice
Name Date 13. 4x 2 16y 2 8x 96y 84 0 14. 49x 2 4y 2 196x 8y 4 0 Chapter l Skills Practice 841
Write an equation in general form for each ellipse. 15. ( x 4)2 144 ( ( x 4)2 144 49 1 16. ( x 1)2 16 49 ) 1 7056 49( x 4) 2 144y 2 7056 (y 2)2 1 81 49( x 2 8x 16) 144y 2 7056 49x 2 392x 784 144y 2 7056 49x 2 144y 2 392x 6272 0 17. ( x 3)2 ( y 2)2 1 18. 49 100 ( x 3)2 ( y 5)2 1 25 4 19. ( x 1)2 49 ( y 1)2 1 20. 25 ( x 2)2 ( y 3)2 1 100 36 842 Chapter l Skills Practice
Name Date Write an equation in standard form for each ellipse. 21. 8 6 4 2 y 8 6 4 2 2 4 6 8 2 x 4 6 8 The center is at (1, 2). The distance between the center and a vertex is 7 units, so a 7. The distance between the center and a co-vertex is 5 units, so b 5. The major axis is horizontal. ( x h)2 The standard form of an ellipse with a horizontal major axis is a ( y k)2 1. 2 2 b ( x 1)2 So, the equation is ( y 2)2 1. 49 25 Chapter l Skills Practice 843
22. 9 6 3 y 9 6 3 3 6 9 3 x 6 9 844 Chapter l Skills Practice
Name Date 23. 8 6 4 2 y 8 6 4 2 2 4 6 8 2 x 4 6 8 Chapter l Skills Practice 845
24. 8 6 4 2 y 8 6 4 2 2 4 6 8 2 x 4 6 8 846 Chapter l Skills Practice
Name Date 25. 8 6 4 2 y 8 6 4 2 2 4 6 8 2 x 4 6 8 Chapter l Skills Practice 847
26. 9 6 3 y 9 6 3 3 6 9 3 x 6 9 848 Chapter l Skills Practice
Skills Practice Skills Practice for Lesson.4 Name Date Whispering Rooms and Laws of Orbits Ellipses and Problem Solving Vocabulary Define the term in your own words. 1. whispering gallery Problem Set Write an equation to model each elliptical room described below. Then create a graph of the shape. Determine how far two people would need to be from the center of the room to hear each other perfectly. 1. The U.S. Capitol building contains a whispering gallery that is 96 feet long and 46 feet wide. The center of the room at eye level is located at the origin of the ellipse. y 20 (0, 23) 10 ( 48, 0) (48, 0) 40 30 20 10 10 20 30 40 x 10 20 The center is (0, 0). The length of the major axis is 96 feet, so a 48. The length of the minor axis is 46 feet, so b 23. The standard form of an ellipse with a horizontal major axis centered at the origin is x2 a 1. 2 2 b So, the equation of the ellipse is x 2 1. 2304 529 Calculate c using the relationship between a, b, and c, c 2 a 2 b 2. c 2 48 2 23 2 2304 529 1775 c 1775 42.13 The two people would each need to stand approximately 42.13 feet from the center of the room. Chapter l Skills Practice 849
2. A whispering gallery has an inner chamber that is 210 feet long and 62 feet high above eye level. The center of the room at eye level is located at the origin of the ellipse. 850 Chapter l Skills Practice
Name Date 3. A whispering gallery has an inner chamber that is 300 feet long and 100 feet high above eye level. The center of the room at eye level is located at the origin of the ellipse. Chapter l Skills Practice 851
4. A whispering gallery has an inner chamber that is 185 feet long and 45 feet high above eye level. The center of the room at eye level is located at the origin of the ellipse. 852 Chapter l Skills Practice
Name Date 5. A whispering gallery has an inner chamber that is 350 feet long and 5 feet high above eye level. The center of the room at eye level is located at the origin of the ellipse. Chapter l Skills Practice 853
6. The Oval Office of the White House is an ellipse with a vertical major axis of 429 inches and a minor axis of 342 inches. 854 Chapter l Skills Practice
Name Date Write an equation to model each elliptical orbit described below. Then create a graph of the orbit. Determine the coordinates of the vertices, the coordinates of the co-vertices, and the coordinates of the foci of the orbit. 7. A space shuttle orbits around the Earth so that the center of the Earth is one focus. Suppose the high point of the orbit is 200 miles above the Earth s surface and the low point is 100 miles above the surface. Assume the diameter of the Earth is about 8000 miles. y 4000 (0, 4149.70) 3000 2000 1000 ( 4150, 0) 4000 2000 1000 (4150, 0) x 2000 4000 2000 3000 4000 (0, 4149.70) For the ellipse modeling the orbit, the major axis has length (8000 100 200) miles, or 8300 miles. The center is located halfway along the major axis, or at (0, 0), and the vertices are ( 4150, 0) and (4150, 0). So, a 4150. Since the center of the Earth is at a focus, the distance from the center to the focus is 50 miles. So, c 50. The coordinates of the foci are ( 50, 0) and (50, 0). Calculate b using the relationship between a, b, and c, b 2 a 2 c 2. b 2 4150 2 50 2 17,222,500 2500 17,220,000 b 17,220,000 4149.70 The coordinates of the co-vertices are (0, 4149.70) and (0, 4149.70). So, the equation is x 2 17,222,500 y 2 1. 17,220,000 Chapter l Skills Practice 855
8. Mars follows an elliptical orbit around the Sun with the Sun at one focus. The widest point of the orbit is 3.0465 AU and the narrowest point of the orbit is 3.0333 AU. An AU is approximately 93,000,000 miles, the distance from the Earth to the Sun. 856 Chapter l Skills Practice
Name Date 9. A star system known as 55 Cancri has a planet that orbits its sun in an elliptical pattern with an eccentricity of 0.24. The length of the vertical major axis is about 5.8 AU. Chapter l Skills Practice 857
10. For the ellipse modeling the orbit of Planet A, the distance between the vertices is approximately 1.45 AU and the distance between the foci is approximately 0.3 AU. 858 Chapter l Skills Practice
Name Date 11. For the ellipse modeling the orbit of Planet B, the distance between the vertices is approximately 1.82 AU and the distance between the foci is approximately 0.38 AU. Chapter l Skills Practice 859
. A moon of Planet N has an elliptical orbit with an eccentricity of 0.751. The length of the major axis is about 11 million kilometers. 860 Chapter l Skills Practice