1. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 1 Homework, Page 682 Find a polar equation for the conic with a focus at the pole and the given eccentricity and directrix. Identify the conic and graph it. 1.

2. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 2 Homework, Page 682 Find a polar equation for the conic with a focus at the pole and the given eccentricity and directrix. Identify the conic and graph it. 5.

3. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 3 Homework, Page 682 Determine the eccentricity, type of conic and directrix. 9.

4. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 4 Homework, Page 682 Determine the eccentricity, type of conic and directrix. 13.

5. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 5 Homework, Page 682

6. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 6 Homework, Page 682 Find a polar equation for an ellipse with a focus at the pole and the given polar coordinates as the endpoints of its major axis.. 21.

7. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 7 Homework, Page 682 Find a polar equation for the hyperbola with a focus at the pole and the given polar coordinates as the endpoints of its transverse axis.. 25.

8. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 8 Homework, Page 682 Find a polar equation for the conic with a focus at the pole. 29.

9. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 9 What you’ll learn about Analyzing Polar Equations of Conics Orbits Revisited … and why You will learn the approach to conics used by astronomers.

10. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 10 Example Analyzing Polar Equations of Conics Graph the conic and find the values of e, a, b, and c. 34.

11. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 11 Example Analyzing Polar Equations of Conics Determine a Cartesian equivalent for the given polar equation. 38.

12. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 12 Example, Writing Cartesian Equations From Polar Equations Use the fact that k = 2p is twice the focal length and half the focal width to determine the Cartesian equation of the parabola whose polar equation is given. 40.

13. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 13 Semimajor Axes and Eccentricities of the Planets

14. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 14 Ellipse with Eccentricity e and Semimajor Axis a

15. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 15 Example Analyzing Orbits of Planets 42.The orbit of the planet Uranus has a semimajor axis of 19.18 AU and an orbital eccentricity of 0.0461. Compute its perihelion and aphelion distances.

16. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 16 Homework Homework Assignment #24 Read Section 8.6 Page 682, Exercises: 33 – 49(Odd), skip 43

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

18. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 18 Quick Review

19. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 19 Quick Review Solutions

20. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 20 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.

21. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 21 The Point P(x,y,z) in Cartesian Space

22. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 22 Features of the Three-Dimensional Cartesian Coordinate System The axes are labeled x, y, and z, and these three coordinate axes form a right-handed coordinate frame. The Cartesian coordinates of a point P in space are an ordered triple, (x, y, z). Pairs of axes determine the coordinate planes. The coordinate planes are the xy-plane, the xz-plane, and the yz-plane and they have equations z = 0, y = 0, and x = 0, respectively. The coordinate planes meet at the origin (0, 0, 0). The coordinate planes divide space into eight regions called octants. The first octant contains all points in space with three positive coordinates.

23. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 23 The Coordinate Planes Divide Space into Eight Octants

24. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 24 Distance Formula (Cartesian Space)

25. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 25 Midpoint Formula (Cartesian Space)

26. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 26 Example Calculating a Distance and Finding a Midpoint

27. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 27 Standard Equation of a Sphere

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

29. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 29 Drawing Lesson (cont’d)

30. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 30 Example Finding the Standard Equation of a Sphere

31. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 31 Equation for a Plane in Cartesian Space

32. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 32 The Vector v =

33. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 33 Vector Relationships in Space

34. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 34 Equations for a Line in Space

35. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 35 Example Finding Equations for a Line