1. Appendices © 2008 Pearson Addison-Wesley. All rights reserved

2. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. Appendices-2 Appendix A Polar Form of Conic Sections Appendix B Rotation of Axes Appendices

3. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. Appendices-3 A Polar Form of Conic Sections

4. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. Appendices-4 A Example 1 Graphing a Conic Section with Equation in Polar Form (cont.) Divide numerator and denominator by 2 to write the equation in standard form. This is the equation of a conic with ep = 3 and e = 2. (See page 1076 in the text.) Thus, p = 1.5. The directrix is vertical and is located 1.5 units to the left of the pole. One focus is at the pole.

5. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. Appendices-5 A Example 1 Graphing a Conic Section with Equation in Polar Form (cont.) Since e > 1, the conic is a hyperbola. (See page 976 in the text.)

6. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. Appendices-6 A Example 1 Graphing a Conic Section with Equation in Polar Form (cont.)

7. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. Appendices-7 A Example 2 Finding a Polar Equation (page 1077) Find the polar equation of a parabola with focus at the pole and horizontal directrix 7 units above the pole. The equation of a conic with focus at the pole and horizontal directrix above the pole is. For a parabola, e = 1. The polar equation is

8. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. Appendices-8 A Example 3 Identifying and Converting From Polar Form to Rectangular Form (page 1077) Identify the type of conic represented by Then convert the equation to rectangular form. Divide numerator and denominator by 2 to write the equation in standard form. This is the equation of a conic with a horizontal directrix, p units above the pole. (See page 1076 in the text.)

9. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. Appendices-9 A Example 3 Identifying and Converting From Polar Form to Rectangular Form (cont.) This is the equation of a conic with ep = –2 and e = 1.5. (See page 1076 in the text.) Thus, p = 1.5. |e| > 1, so the conic is a hyperbola. The foci are at (0, 0) and

10. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. Appendices-10 A Example 3 Identifying and Converting From Polar Form to Rectangular Form (cont.) Use the equations to convert the equation to rectangular form.

11. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. Appendices-11 Rotation of Axes B Derivation of Rotation Equations ▪ Applying a Rotation Equation

12. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. Appendices-12 B Example 1 Finding an Equation After a Rotation (page 1080) The equation of a curve is Find the resulting equation if the axes are rotated 45°. Graph the equation. The rotation equations are and

13. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. Appendices-13 B Example 1 Finding an Equation After a Rotation (cont.) Substitute the rotation equations into the original equation.

14. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. Appendices-14 B Example 1 Finding an Equation After a Rotation (cont.) This is the equation of an ellipse with x′-intercepts and y′-intercepts

15. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. Appendices-15 B Example 2 Rotating and Graphing (page 1081) Remove the xy-term from by performing a suitable rotation, and graph the equation. A = 1, B = 1, C = 1, so The rotation equations are and

16. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. Appendices-16 B Example 2 Rotating and Graphing (cont.) Substitute the rotation equations into the original equation.

17. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. Appendices-17 B Example 2 Rotating and Graphing (cont.) This is the equation of an ellipse with x′-intercepts and and y′-intercepts