1. Chapter 9 Conic Sections and Analytic Geometry Copyright © 2014, 2010, 2007 Pearson Education, Inc. 1 9.4 Rotation of Axes

2. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 2 Objectives: Identify conics without completing the square. Use rotation of axes formulas. Write equations of rotated conics in standard form. Identify conics without rotating axes.

3. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 3 Identifying Conic Sections without Completing the Square Conic sections can be represented both geometrically (as intersecting planes and cones) and algebraically. The equations of the conic sections we have considered in the first three sections of this chapter can be expressed in the form in which A and C are not both zero. We can identify a conic section without completing the square by comparing the values of A and C.

4. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 4 Identifying a Conic Section without Completing the Square (continued)

5. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 5 Example: Identifying a Conic Section without Completing the Square Identify the graph of the following nondegenerate conic section: Because A is not equal to C and AC is positive, the graph of the equation is an ellipse.

6. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 6 Example: Identifying a Conic Section without Completing the Square Identify the graph of the following nondegenerate conic section: Because A and C are equal, the graph of the equation is a circle.

7. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 7 Example: Identifying a Conic Section without Completing the Square Identify the graph of the following nondegenerate conic section: Because AC = 0, the graph of the equation is a parabola.

8. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 8 Example: Identifying a Conic Section without Completing the Square Identify the graph of the following nondegenerate conic section: Because AC is negative, the graph of the equation is an hyperbola.

9. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 9 Rotation of Axes Except for degenerate cases, the general second-degree equation represents one of the conic sections. Due to the xy-term in the equation, these conic sections are rotated in such a way that their axes are no longer parallel to the x- and y-axes. To reduce these equations to forms of the conic sections with which we are already familiar, we use a procedure called rotation of axes.

10. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 10 Rotation of Axes Formula

11. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 11 Example: Rotating Axes Write the equation xy = 2 in terms of a rotated x′y′-system if the angle of rotation from the x-axis to the x′-axis is 45°. Express the equation in standard form. Use the rotated system to graph xy = 2.

12. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 12 Example: Rotating Axes (continued) Write the equation xy = 2 in terms of a rotated x′y′-system if the angle of rotation from the x-axis to the x′-axis is 45°. Express the equation in standard form. Use the rotated system to graph xy = 2.

13. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 13 Example: Rotating Axes (continued) Write the equation xy = 2 in terms of a rotated x′y′-system if the angle of rotation from the x-axis to the x′-axis is 45°. Express the equation in standard form. Use the rotated system to graph xy = 2.

14. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 14 Example: Rotating Axes (continued) Write the equation xy = 2 in terms of a rotated x′y′-system if the angle of rotation from the x-axis to the x′-axis is 45°. Express the equation in standard form. Use the rotated system to graph xy = 2. The graph of xy = 2 or vertex (2, 0) vertex (–2, 0)

15. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 15 Using Rotations to Transform Equations with xy-Terms to Standard Equations of Conic Sections A rotation of axes through an appropriate angle can transform the equation to one of the standard forms of the conic sections in x′ and y′ in which no x′y′-term appears.

16. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 16 Amount of Rotation Formula

17. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 17 Writing the Equation of a Rotated Conic in Standard Form

18. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 18 Example: Writing the Equation of a Rotated Conic Section in Standard Form Rewrite the equation in a rotated x′y′-system without an x′y′-term. Express the equation in the standard form of a conic section. Graph the conic section in the rotated system. Step 1 Use the given equation to find

19. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 19 Example: Writing the Equation of a Rotated Conic Section in Standard Form (continued) Rewrite the equation in a rotated x′y′-system without an x′y′-term. Express the equation in the standard form of a conic section. Graph the conic section in the rotated system. Step 2 Use the expression for to determine the angle of rotation.

20. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 20 Example: Writing the Equation of a Rotated Conic Section in Standard Form (continued) Rewrite the equation in a rotated x′y′-system without an x′y′-term. Express the equation in the standard form of a conic section. Graph the conic section in the rotated system. Step 3 Substitute in the rotation formulas and simplify.

21. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 21 Example: Writing the Equation of a Rotated Conic Section in Standard Form (continued) Rewrite the equation in a rotated x′y′-system without an x′y′-term. Express the equation in the standard form of a conic section. Graph the conic section in the rotated system. Step 3 (cont) Substitute in the rotation formulas and simplify.

22. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 22 Example: Writing the Equation of a Rotated Conic Section in Standard Form (continued) Step 4 Substitute the expressions for x and y from the rotation formulas in the given equation and simplify.

23. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 23 Example: Writing the Equation of a Rotated Conic Section in Standard Form (continued) Step 4 (cont) Substitute the expressions for x and y from the rotation formulas in the given equation and simplify.

24. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 24 Example: Writing the Equation of a Rotated Conic Section in Standard Form (continued) Step 4 (cont) Substitute the expressions for x and y from the rotation formulas in the given equation and simplify.

25. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 25 Example: Writing the Equation of a Rotated Conic Section in Standard Form (continued) Step 4 (cont) Substitute the expressions for x and y from the rotation formulas in the given equation and simplify. Step 5 Write the equation involving x′ and y′ in standard form.

26. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 26 Example: Writing the Equation of a Rotated Conic Section in Standard Form (continued) Rewrite the equation in a rotated x′y′-system without an x′y′-term. Express the equation in the standard form of a conic section. Graph the conic section in the rotated system. Graph of or Major axis y′

27. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 27 Identifying Conic Sections without Rotating Axes

28. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 28 Example: Identifying a Conic Section without Rotating Axes Identify the graph of Because the equation is a parabola.