1. PreCalculus 9-R Unit 9 – Analytic Geometry Review Problems

2. Find the focus, directrix, and focal diameter of the parabola, and sketch its graph. 1

3. Review Problems Find the focus, directrix, and focal diameter of the parabola, and sketch its graph. 2

4. Review Problems Find the focus of the parabola y 2 = 8x (2, 0) 3

5. Review Problems Find the directrix of the parabola x 2 = 4y y = –1 4

6. Review Problems Find an equation for the parabola that has its vertex at the origin and focus F (0, 3). x 2 = 12y 5

7. Review Problems Find an equation for the parabola whose directrix has y-intercept 7. x 2 = –28y 6

8. Review Problems Find an equation of the parabola whose graph is shown. x 2 = 16y 7

9. Review Problems A lamp with a parabolic reflector is shown in the figure. The bulb is placed at the focus, and the focal diameter is 4 cm. Find the diameter d(C, D) of the opening 15 cm from the vertex. ( a = 2, b = 15 ) 15 cm 8

10. Review Problems Find the vertices and foci of the ellipse, and sketch the graph. 9

11. Review Problems Find the intersection points of the pair of ellipses. Sketch the graphs of the pair of equations on the same coordinate axes and label the points of intersection. 10

12. Review Problems Find the vertices of the ellipse. (–10, 0) and (10, 0) 11

13. Review Problems foci, major axis 2, minor axis 12

14. Review Problems Find an equation for the ellipse whose graph is shown. 13

15. Review Problems Find an equation for the ellipse that satisfies the given conditions. foci (  11, 0 ), vertices (  12, 0 ) 14

16. Review Problems Find an equation for the ellipse that satisfies the given conditions. Length of major axis 4, Foci on y-axis, ellipse passes through the point (, 0 ) 15

17. Review Problems A "sunburst" window above a doorway is constructed in the shape of the top half of an ellipse, as shown in the figure. The window is 30 in tall at its highest point and 60 in wide at the bottom. Find the height of the window 10 in from the center of the base. 16

18. Review Problems Find the vertices, foci, and asymptotes of the hyperbola, and sketch its graph. 17

19. Review Problems Find the vertices, foci, and asymptotes of the hyperbola, and sketch its graph. 18

20. Review Problems Find the vertices, foci, and asymptotes of the hyperbola, and sketch its graph. 19

21. Review Problems Find the vertices of the hyperbola. (  4, 0) 20

22. Review Problems Find the foci of the hyperbola. 4x 2 – 25y 2 = 100 21

23. Review Problems Find the asymptotes of the hyperbola. 9y 2 – x 2 = 1 22

24. Review Problems Find the equation for the hyperbola whose graph is shown. The foci of the hyperbola are F 1 (–5, 0) and F 2 (5, 0). 23

25. Review Problems Find the equation for the hyperbola with foci (  5, 0) and vertices (  4, 0). 24

26. Review Problems Find the equation for the hyperbola with foci (0,  8) and asymptotes 25

27. Review Problems Find the equation for the hyperbola with the foci (  10, 0) with transverse axis of length 16. 26

28. Review Problems Find the center, foci, and vertices of the ellipse, and determine the lengths of the major and minor axes. Then sketch the graph. The length of the major axis is 10, and the length of the minor axis is 6 27

29. Review Problems Find the vertex, focus, and directrix of the parabola, and sketch the graph. 28

30. Review Problems Find the center, foci, vertices, and asymptotes of the hyperbola. Then sketch the graph. The center is (–3, 1), the foci are, the vertices are (–3, 4) and (–3, –2), and the asymptotes are and 29

31. Review Problems Complete the square to determine whether the equation represents an ellipse, a parabola, a hyperbola, or a degenerate conic. If the graph is an ellipse, find the center, foci, vertices, and lengths of the major and minor axes. If it is a parabola, find the vertex, focus, and directrix. If it is a hyperbola, find the center, foci, vertices, and asymptotes. Then sketch the graph of the equation. 30

32. Review Problems Complete the square to determine whether the equation represents an ellipse, a parabola, a hyperbola, or a degenerate conic. If the graph is an ellipse, find the center, foci, vertices, and lengths of the major and minor axes. If it is a parabola, find the vertex, focus, and directrix. If it is a hyperbola, find the center, foci, vertices, and asymptotes. Then sketch the graph of the equation. The length of major axis is 8, and the length of minor axis is 6. 31

33. Review Problems Find the lengths of the major and minor axes of the ellipse. major axis 16, minor axis 4 32

34. Review Problems Find the vertex and focus of the parabola. –5(x + 8) 2 = y vertex (–8, 0), focus 33

35. Review Problems Find the center and vertices of the hyperbola. (x – 7) 2 – (y + 6) 2 = 1 center (7, –6), vertices (8, –6), (6, –6) 34

36. Review Problems Find an equation for the conic whose graph is shown. y 2 = 4.8(x + 2) 35

37. Review Problems Find an equation for the conic whose graph is shown. 36

38. Review Problems Find the center and lengths of the major and minor axes of the ellipse. 36y 2 + 9x 2 – 108x – 72y + 36 = 0 center (6, 1), major axis 12, minor axis 6 37

39. Review Problems Find the vertex and directrix of the parabola. x 2 – 4x – 28y – 248 = 0 vertex (2, –9), directrix y = –16 38

40. Review Problems Sketch the graph. 39

41. Review Problems Determine the XY-coordinates of the point if the coordinate axes are rotated through the angle 40

42. Review Problems Determine the XY-coordinates of the point if the coordinate axes are rotated through the angle 41

43. Review Problems Determine the equation of the conic in XY-coordinates when the coordinate axes are rotated through the angle 42

44. Review Problems Use the discriminant to determine whether the graph of the equation is a parabola, an ellipse, or a hyperbola. hyperbola 43

45. Review Problems Use a rotation of axes to eliminate the xy-term in the equation 44

46. Review Problems Use a rotation of axes to eliminate the xy-term in the equation X 2 – 4Y 2 = 8 45

47. Review Problems Use a rotation of axes to eliminate the xy-term in the equation 46

48. Review Problems Use a rotation of axes to eliminate the xy-term in the equation 47

49. Review Problems A pair of parametric equations is given. Sketch the curve represented by the parametric equations. 48

50. Review Problems A pair of parametric equations is given. Sketch the curve represented by the parametric equations. 49

51. Review Problems Sketch the curve given by the parametric equations 50

52. Review Problems Use a graphing device to draw the curve represented by the parametric equations. 51

53. Review Problems Use a graphing device to draw the curve represented by the parametric equations. 52

54. Review Problems Use a graphing device to draw the curve represented by the parametric equations. 53

55. Review Problems 54 Plot the point that has the polar coordinates.

56. Review Problems 55 Plot the point that has the polar coordinates

57. Review Problems 56 Sketch the graph of the polar equation.

58. Review Problems 57 Sketch the graph of the polar equation.

59. Review Problems 58 Sketch the graph of the polar equation.

60. Review Problems 59 Sketch the graph of the polar equation.

61. Review Problems 60 Sketch the graph of the polar equation.

62. Review Problems 61 Sketch the graph of the polar equation.

63. Review Problems 62 Sketch the graph of the polar equation.

64. Review Problems 63 Graph the polar equation.

65. Review Problems 64 Sketch a graph of the rectangular equation.

66. Review Problems 65 Sketch a graph of the rectangular equation.

67. Review Problems 66 Graph the polar equation.

68. Review Problems 67 Graph the polar equation.

69. Review Problems 68 Test the polar equation for symmetry. The equation is symmetric with respect to the line. Test the polar equation for symmetry. The equation is symmetric with respect to the polar axis. The equation is symmetric with respect to the pole. The equation is symmetric with respect to the line.

70. Review Problems 69 Give two polar coordinate representations of the point one with and the other with and A point is graphed in rectangular form. Find polar coordinates for the point, with and.

71. Review Problems 70 A point is graphed in polar form. Find its rectangular coordinates. Find the rectangular coordinates for the point whose polar coordinates are (0, 1)

72. Review Problems 71 Find the rectangular coordinates for the point whose polar coordinates are Convert the rectangular coordinates to polar coordinates with and

73. Review Problems 72 Convert the equation to polar form

74. Review Problems 73 Convert the polar equation to rectangular coordinates.

75. ANSWERS 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. (2, 0) y = –1 x 2 = 12y x 2 = –28y x 2 = 16y 15 cm (–10, 0) and (10, 0) foci, major axis 2, minor axis

76. ANSWERS 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. (-8,8),(-6,8) 31. 32. 33. 34. 35. 36. 37. 38. 39. The length of the major axis is 10, and the length of the minor axis is 6 (–3, 1), major axis is 8, minor axis is 6. major axis 16, minor axis 4 vertex (–8, 0), focus center (7, –6), vertices (8, –6), (6, –6) y 2 = 4.8(x + 2) center (6, 1), major axis 12, minor axis 6 vertex (2, –9), directrix y = –16 (  4, 0)

77. ANSWERS 40. 41. 42. 43. 44. 45. 46. 47. 48. 49. 50. 51. 52. 53. hyperbola X 2 – 4Y 2 = 8

78. ANSWERS 54. 55. 56. 57. 58. 59. 60. 61. 62. 63.

79. ANSWERS 64. 65. 66. 67. 68. 69. 70. 71. 72. 73. The equation is symmetric with respect to the line The equation is symmetric with respect to the polar axis. The equation is symmetric with respect to the pole. The equation is symmetric with respect to the line (0, 1)