Skills Practice Skills Practice for Lesson 12.1

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1 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 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

2 Problem Set Given the equation of each ellipse in standard form, write the equation of the ellipse in general form. 1. x x ( x ) 1 9x 2 16y x 2 16y x x x x Given the equation of each ellipse in general form, write the equation of the ellipse in standard form x 2 25y x 2 9y x 2 25y x 2 25y x x Chapter l Skills Practice

3 Name Date 9. 81x 2 16y x 2 9y x 2 2y x 2 16y 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 x 2 semi-major 4 axis minor axis 6 vertex 8 Chapter l Skills Practice 813

4 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

5 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 x y (0, 3) 2 ( 8, 0) (8, 0) (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 c The coordinates of the foci are (7.42, 0), ( 7.42, 0). Chapter l Skills Practice 815

6 21. x x Chapter l Skills Practice

7 Name Date 23. x x Chapter l Skills Practice 817

8 Given the equation of an ellipse in standard form, calculate the eccentricity of each ellipse. Then sketch the ellipse. 25. x x y x The major axis is along the y-axis. a 6 and b 5 c 2 a 2 b 2 c c e c a Chapter l Skills Practice

9 Name Date 27. x x Chapter l Skills Practice 819

10 29. x x Chapter l Skills Practice

11 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 x y 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

12 33. x x Chapter l Skills Practice

13 Name Date 35. x x Chapter l Skills Practice 823

14 824 Chapter l Skills Practice

15 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 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

16 Identify the center of each ellipse. Then graph the ellipse. 7. (x 1) ( x 2)2 25 ( y 1) y (1, 0) x (1, 0) 9. (x 4)2 ( y 3) ( x 2)2 ( y 2) Chapter l Skills Practice

17 Name Date 11. (x 3)2 ( y 1) ( x 4)2 ( y 1) 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) y ( x 2)2 ( y 4) (1, 5) x (10.75, 2) 2 (11, 2) ( 8.75, 2) (1, 2) (13, 2) (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

18 15. (x 1)2 ( y 3) ( x 2)2 ( y 5) Chapter l Skills Practice

19 Name Date 17. (x 4)2 ( y 3) ( x 3)2 ( y 1) Chapter l Skills Practice 829

20 830 Chapter l Skills Practice

21 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 x 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 b So, the equation is x Chapter l Skills Practice 831

22 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

23 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

24 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

25 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

26 6. An ellipse with vertices at (1, 3) and ( 9, 3) and one focus at (0, 3). 836 Chapter l Skills Practice

27 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

28 8. An ellipse centered at ( 5, 3) with a vertical major axis and eccentricity of Chapter l Skills Practice

29 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 x 2 25y x 2 16y 2 36x 32y y x x 2 25y x 2 25y x x 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

30 11. 25x 2 16y 2 150x 32y x 2 49y 2 200x 196y Chapter l Skills Practice

31 Name Date 13. 4x 2 16y 2 8x 96y x 2 4y 2 196x 8y 4 0 Chapter l Skills Practice 841

32 Write an equation in general form for each ellipse. 15. ( x 4)2 144 ( ( x 4) ( x 1) ) ( x 4) 2 144y (y 2) ( x 2 8x 16) 144y x 2 392x y x 2 144y 2 392x ( x 3)2 ( y 2) ( x 3)2 ( y 5) ( x 1)2 49 ( y 1) ( x 2)2 ( y 3) Chapter l Skills Practice

33 Name Date Write an equation in standard form for each ellipse y x 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) b ( x 1)2 So, the equation is ( y 2) Chapter l Skills Practice 843

34 y x Chapter l Skills Practice

35 Name Date y x Chapter l Skills Practice 845

36 y x Chapter l Skills Practice

37 Name Date y x Chapter l Skills Practice 847

38 y x Chapter l Skills Practice

39 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) x 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 b So, the equation of the ellipse is x Calculate c using the relationship between a, b, and c, c 2 a 2 b 2. c c The two people would each need to stand approximately feet from the center of the room. Chapter l Skills Practice 849

40 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

41 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

42 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

43 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

44 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

45 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, ) ( 4150, 0) (4150, 0) x (0, ) For the ellipse modeling the orbit, the major axis has length ( ) 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 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 ,222, ,220,000 b 17,220, The coordinates of the co-vertices are (0, ) and (0, ). So, the equation is x 2 17,222,500 y ,220,000 Chapter l Skills Practice 855

46 8. Mars follows an elliptical orbit around the Sun with the Sun at one focus. The widest point of the orbit is AU and the narrowest point of the orbit is AU. An AU is approximately 93,000,000 miles, the distance from the Earth to the Sun. 856 Chapter l Skills Practice

47 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 The length of the vertical major axis is about 5.8 AU. Chapter l Skills Practice 857

48 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

49 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

50 . A moon of Planet N has an elliptical orbit with an eccentricity of The length of the major axis is about 11 million kilometers. 860 Chapter l Skills Practice

Notes 10-3: Ellipses

Notes 10-3: Ellipses I. Ellipse- Definition and Vocab An ellipse is the set of points P(x, y) in a plane such that the sum of the distances from any point P on the ellipse to two fixed points F 1 and F