1. Slide 8- 1

2. Chapter 8 Analytic Geometry in Two and Three Dimensions

3. 8.1 Conic Sections and Parabolas

4. Slide 8- 4 Quick Review

5. Slide 8- 5 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.

6. Slide 8- 6 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.

7. Slide 8- 7 A Right Circular Cone (of two nappes)

8. Slide 8- 8 Conic Sections and Degenerate Conic Sections

9. Slide 8- 9 Conic Sections and Degenerate Conic Sections (cont’d) Animation

10. Slide 8- 10 Second-Degree (Quadratic) Equations in Two Variables

11. Slide 8- 11 Structure of a Parabola

12. Slide 8- 12 Graphs of x 2 =4py

13. Slide 8- 13 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|

14. Slide 8- 14 Graphs of y 2 = 4px

15. Slide 8- 15 Example Finding an Equation of a Parabola

16. Slide 8- 16 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|

17. Slide 8- 17 Example Finding an Equation of a Parabola

18. 8.2 Ellipses

19. Slide 8- 19 Quick Review

20. Slide 8- 20 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.

21. Slide 8- 21 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.

22. Slide 8- 22 Key Points on the Focal Axis of an Ellipse

23. Slide 8- 23 Ellipse with Center (0,0)

24. Slide 8- 24 Pythagorean Relation

25. Slide 8- 25 Example Finding the Vertices and Foci of an Ellipse

26. Slide 8- 26 Example Finding an Equation of an Ellipse

27. Slide 8- 27 Ellipse with Center (h,k)

28. Slide 8- 28 Ellipse with Center (h,k)

29. Slide 8- 29 Example Locating Key Points of an Ellipse

30. Slide 8- 30 Elliptical Orbits Around the Sun

31. Slide 8- 31 Eccentricity of an Ellipse

32. 8.3 Hyperbolas

33. Slide 8- 33 Quick Review

34. Slide 8- 34 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.

35. Slide 8- 35 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.

36. Slide 8- 36 Hyperbola

37. Slide 8- 37 Hyperbola

38. Slide 8- 38 Hyperbola with Center (0,0)

39. Slide 8- 39 Hyperbola Centered at (0,0)

40. Slide 8- 40 Example Finding the Vertices and Foci of a Hyperbola 1

41. Slide 8- 41 Example Finding an Equation of a Hyperbola

42. Slide 8- 42 Hyperbola with Center (h,k)

43. Slide 8- 43 Hyperbola with Center (h,k)

44. Slide 8- 44 Example Locating Key Points of a Hyperbola

45. Slide 8- 45 Eccentricity of a Hyperbola

46. 8.4 Translations and Rotations of Axes

47. Slide 8- 47 Quick Review

48. Slide 8- 48 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.

49. Slide 8- 49 Translation-of-Axes Formulas

50. Slide 8- 50 Example Translation Formula

51. Slide 8- 51 Rotation-of-Axes Formulas

52. Slide 8- 52 Rotation of Cartesian Coordinate Axes

53. Slide 8- 53 Example Rotation of Axes

54. Slide 8- 54 Example Rotation of Axes

55. Slide 8- 55 Coefficients for a Conic in a Rotated System

56. Slide 8- 56 Angle of Rotation to Eliminate the Cross- Product Term

57. Slide 8- 57 Discriminant Test

58. Slide 8- 58 Conics and the Equation Ax 2 +Bxy+Cy 2 +Dx+Ey+F=0

59. 8.5 Polar Equations of Conics

60. Slide 8- 60 Quick Review

61. Slide 8- 61 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.

62. Slide 8- 62 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.)

63. Slide 8- 63 Focus-Directrix Eccentricity Relationship

64. Slide 8- 64 The Geometric Structure of a Conic Section

65. Slide 8- 65 A Conic Section in the Polar Plane

66. Slide 8- 66 Three Types of Conics for r = ke/(1+ecosθ)

67. Slide 8- 67 Polar Equations for Conics

68. Slide 8- 68 Example Writing Polar Equations of Conics

69. Slide 8- 69 Example Identifying Conics from Their Polar Equations

70. Slide 8- 70 Semimajor Axes and Eccentricities of the Planets

71. Slide 8- 71 Ellipse with Eccentricity e and Semimajor Axis a

72. 8.6 Three-Dimensional Cartesian Coordinate System

73. Slide 8- 73 Quick Review

74. Slide 8- 74 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.

75. Slide 8- 75 The Point P(x,y,z) in Cartesian Space

76. Slide 8- 76 The Coordinate Planes Divide Space into Eight Octants

77. Slide 8- 77 Distance Formula (Cartesian Space)

78. Slide 8- 78 Midpoint Formula (Cartesian Space)

79. Slide 8- 79 Example Calculating a Distance and Finding a Midpoint

80. Slide 8- 80 Standard Equation of a Sphere

81. Slide 8- 81 Drawing Lesson

82. Slide 8- 82 Drawing Lesson (cont’d)

83. Slide 8- 83 Example Finding the Standard Equation of a Sphere

84. Slide 8- 84 Equation for a Plane in Cartesian Space

85. Slide 8- 85 The Vector v =

86. Slide 8- 86 Vector Relationships in Space

87. Slide 8- 87 Equations for a Line in Space

88. Slide 8- 88 Example Finding Equations for a Line

89. Slide 8- 89 Chapter Test

90. Slide 8- 90 Chapter Test

91. Slide 8- 91 Chapter Test Solutions

92. Slide 8- 92 Chapter Test Solutions