1. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 1

2. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Chapter 8 Analytic Geometry in Two and Three Dimensions

3. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 8.1 Conic Sections and Parabolas

4. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 4 Quick Review

5. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 5 Quick Review Solutions

6. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 6 What you’ll learn about Conic Sections Geometry of a Parabola Translations of Parabolas Reflective Property of a Parabola … and why Conic sections are the paths of nature: Any free-moving object in a gravitational field follows the path of a conic section.

7. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 7 Parabola A parabola is the set of all points in a plane equidistant from a particular line (the directrix) and a particular point (the focus) in the plane.

8. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 8 A Right Circular Cone (of two nappes)

9. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 9 Conic Sections and Degenerate Conic Sections

10. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 10 Conic Sections and Degenerate Conic Sections (cont’d)

11. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 11 Second-Degree (Quadratic) Equations in Two Variables

12. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 12 Structure of a Parabola

13. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 13 Graphs of x 2 =4py

14. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 14 Parabolas with Vertex (0,0) Standard equationx 2 = 4pyy 2 = 4px Opens Upward or To the right or to the downward left Focus(0,p)(p,0) Directrixy = -px = -p Axisy-axisx-axis Focal lengthpp Focal width|4p||4p|

15. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 15 Graphs of y 2 = 4px

16. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 16 Example Finding an Equation of a Parabola

17. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 17 Example Finding an Equation of a Parabola

18. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 18 Parabolas with Vertex (h,k) Standard equation (x-h) 2 = 4p(y-k)(y-k) 2 = 4p(x-h) Opens Upward or To the right or to the downward left Focus(h,k+p)(h+p,k) Directrixy = k-px = h-p Axisx = hy = k Focal lengthpp Focal width|4p||4p|

19. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 19 Example Finding an Equation of a Parabola

20. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 20 Example Finding an Equation of a Parabola

21. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 8.2 Ellipses

22. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 22 Quick Review

23. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 23 Quick Review Solutions

24. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 24 What you’ll learn about Geometry of an Ellipse Translations of Ellipses Orbits and Eccentricity Reflective Property of an Ellipse … and why Ellipses are the paths of planets and comets around the Sun, or of moons around planets.

25. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 25 Ellipse An ellipse is the set of all points in a plane whose distance from two fixed points in the plane have a constant sum. The fixed points are the foci (plural of focus) of the ellipse. The line through the foci is the focal axis. The point on the focal axis midway between the foci is the center. The points where the ellipse intersects its axis are the vertices of the ellipse.

26. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 26 Key Points on the Focal Axis of an Ellipse

27. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 27 Ellipse with Center (0,0)

28. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 28 Pythagorean Relation

29. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 29 Example Finding the Vertices and Foci of an Ellipse

30. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 30 Example Finding the Vertices and Foci of an Ellipse

31. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 31 Example Finding an Equation of an Ellipse

32. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 32 Example Finding an Equation of an Ellipse

33. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 33 Ellipse with Center (h,k)

34. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 34 Ellipse with Center (h,k)

35. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 35 Example Locating Key Points of an Ellipse

36. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 36 Example Locating Key Points of an Ellipse

37. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 37 Elliptical Orbits Around the Sun

38. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 38 Eccentricity of an Ellipse

39. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 8.3 Hyperbolas

40. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 40 Quick Review

41. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 41 Quick Review Solutions

42. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 42 What you’ll learn about Geometry of a Hyperbola Translations of Hyperbolas Eccentricity and Orbits Reflective Property of a Hyperbola Long-Range Navigation … and why The hyperbola is the least known conic section, yet it is used astronomy, optics, and navigation.

43. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 43 Hyperbola A hyperbola is the set of all points in a plane whose distances from two fixed points in the plane have a constant difference. The fixed points are the foci of the hyperbola. The line through the foci is the focal axis. The point on the focal axis midway between the foci is the center. The points where the hyperbola intersects its focal axis are the vertices of the hyperbola.

44. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 44 Hyperbola

45. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 45 Hyperbola

46. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 46 Hyperbola with Center (0,0)

47. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 47 Hyperbola Centered at (0,0)

48. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 48 Example Finding the Vertices and Foci of a Hyperbola

49. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 49 Example Finding the Vertices and Foci of a Hyperbola

50. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 50 Example Finding an Equation of a Hyperbola

51. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 51 Example Finding an Equation of a Hyperbola

52. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 52 Hyperbola with Center (h,k)

53. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 53 Hyperbola with Center (h,k)

54. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 54 Example Locating Key Points of a Hyperbola

55. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 55 Example Locating Key Points of a Hyperbola

56. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 56 Eccentricity of a Hyperbola

57. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 8.4 Translations and Rotations of Axes

58. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 58 Quick Review

59. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 59 Quick Review Solutions

60. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 60 What you’ll learn about Second-Degree Equations in Two Variables Translating Axes versus Translating Graphs Rotation of Axes Discriminant Test … and why You will see ellipses, hyperbolas, and parabolas as members of the family of conic sections rather than as separate types of curves.

61. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 61 Translation-of-Axes Formulas

62. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 62 Example Translation Formula

63. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 63 Example Translation Formula

64. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 64 Rotation-of-Axes Formulas

65. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 65 Rotation of Cartesian Coordinate Axes

66. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 66 Example Rotation of Axes

67. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 67 Example Rotation of Axes

68. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 68 Coefficients for a Conic in a Rotated System

69. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 69 Angle of Rotation to Eliminate the Cross- Product Term

70. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 70 Discriminant Test

71. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 71 Conics and the Equation Ax 2 +Bxy+Cy 2 +Dx+Ey+F=0

72. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 8.5 Polar Equations of Conics

73. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 73 Quick Review

74. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 74 Quick Review Solutions

75. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 75 What you’ll learn about Eccentricity Revisited Writing Polar Equations for Conics Analyzing Polar Equations of Conics Orbits Revisited … and why You will learn the approach to conics used by astronomers.

76. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 76 Focus-Directrix Definition Conic Section A conic section is the set of all points in a plane whose distances from a particular point (the focus) and a particular line (the directrix) in the plane have a constant ratio. (We assume that the focus does not lie on the directrix.)

77. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 77 Focus-Directrix Eccentricity Relationship

78. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 78 The Geometric Structure of a Conic Section

79. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 79 A Conic Section in the Polar Plane

80. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 80 Three Types of Conics for r = ke/(1+ecosθ)

81. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 81 Polar Equations for Conics

82. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 82 Example Writing Polar Equations of Conics

83. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 83 Example Writing Polar Equations of Conics

84. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 84 Example Identifying Conics from Their Polar Equations

85. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 85 Example Identifying Conics from Their Polar Equations

86. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 86 Semimajor Axes and Eccentricities of the Planets

87. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 87 Ellipse with Eccentricity e and Semimajor Axis a

88. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 8.6 Three-Dimensional Cartesian Coordinate System

89. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 89 Quick Review

90. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 90 Quick Review Solutions

91. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 91 What you’ll learn about Three-Dimensional Cartesian Coordinates Distances and Midpoint Formula Equation of a Sphere Planes and Other Surfaces Vectors in Space Lines in Space … and why This is the analytic geometry of our physical world.

92. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 92 The Point P(x,y,z) in Cartesian Space

93. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 93 The Coordinate Planes Divide Space into Eight Octants

94. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 94 Distance Formula (Cartesian Space)

95. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 95 Midpoint Formula (Cartesian Space)

96. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 96 Example Calculating a Distance and Finding a Midpoint

97. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 97 Example Calculating a Distance and Finding a Midpoint

98. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 98 Standard Equation of a Sphere

99. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 99 Drawing Lesson

100. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 100 Drawing Lesson (cont’d)

101. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 101 Example Finding the Standard Equation of a Sphere

102. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 102 Example Finding the Standard Equation of a Sphere

103. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 103 Equation for a Plane in Cartesian Space

104. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 104 The Vector v =

105. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 105 Vector Relationships in Space

106. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 106 Equations for a Line in Space

107. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 107 Example Finding Equations for a Line

108. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 108 Example Finding Equations for a Line

109. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 109 Chapter Test

110. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 110 Chapter Test

111. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 111 Chapter Test Solutions

112. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 112 Chapter Test Solutions