1. Copyright © 2007 Pearson Education, Inc. Slide 6-1

2. Copyright © 2007 Pearson Education, Inc. Slide 6-2 Chapter 6: Analytic Geometry 6.1Circles and Parabolas 6.2Ellipses and Hyperbolas 6.3Summary of the Conic Sections 6.4Parametric Equations

3. Copyright © 2007 Pearson Education, Inc. Slide 6-3 Conic sections presented in this chapter are of the form where either A or C must be nonzero. 6.3 Summary of the Conic Sections Conic SectionCharacteristic Example Parabola Circle Ellipse Hyperbola Either A = 0 or C = 0, but not both. y = x 2 x = 3y 2 + 2y – 4

4. Copyright © 2007 Pearson Education, Inc. Slide 6-4 6.3 Summary of the Conic Sections

5. Copyright © 2007 Pearson Education, Inc. Slide 6-5 6.3 Summary of the Conic Sections Last 3 rows of the Table on pg 6-52

6. Copyright © 2007 Pearson Education, Inc. Slide 6-6 6.3 Determining Type of Conic Section To recognize the type of conic section, we may need to transform the equation. Example Decide on the type of conic section represented by each equation. (a) (b) (c) (d)

7. Copyright © 2007 Pearson Education, Inc. Slide 6-7 6.3 Determining Type of Conic Section Solution (a) This equation represents a hyperbola centered at the origin.

8. Copyright © 2007 Pearson Education, Inc. Slide 6-8 6.3 Determining Type of Conic Section (b)Coefficients of the x 2 - and y 2 -terms are unequal and both positive. This equation may be an ellipse. Complete the square on x and y. This is an ellipse centered at (2, –3).

9. Copyright © 2007 Pearson Education, Inc. Slide 6-9 6.3 Determining Type of Conic Section (c)Coefficients of the x 2 - and y 2 -terms are both 1. This equation may be a circle. Complete the square on x and y. This equation is a circle with radius 0; that is, the point (4, –5).

10. Copyright © 2007 Pearson Education, Inc. Slide 6-10 6.3 Determining Type of Conic Section (d)Since only one variable is squared, x 2, the equation represents a parabola. Solve for y (the variable that is not squared) and complete the square on x (the squared variable). The parabola has vertex (3, 2) and opens downward.

11. Copyright © 2007 Pearson Education, Inc. Slide 6-11 6.3 Eccentricity The constant ratio is called the eccentricity of the conic, written e. A conic is the set of all points P(x, y) in a plane such that the ratio of the distance from P to a fixed point and the distance from P to a fixed line is constant.

12. Copyright © 2007 Pearson Education, Inc. Slide 6-12 6.3 Eccentricity of Ellipses and Hyperbolas Ellipses and hyperbolas have eccentricity where c is the distance from the center to a focus. For ellipses, a 2 > b 2 and Note that ellipses with eccentricity close to 0 have a circular shape because b  a as c  0.

13. Copyright © 2007 Pearson Education, Inc. Slide 6-13 6.3 Finding Eccentricity of an Ellipse ExampleFind the eccentricity of SolutionSince 16 > 9, let a 2 = 16, giving a = 4.

14. Copyright © 2007 Pearson Education, Inc. Slide 6-14 6.3 Figures Comparing Different Eccentricities of Ellipses and Hyperbolas

15. Copyright © 2007 Pearson Education, Inc. Slide 6-15 6.3 Finding Equations of Conics using Eccentricity ExampleFind an equation for each conic with center at the origin. (a)Focus at (3, 0) and eccentricity 2 (b)Vertex at (0, –8) and e = Solution (a)e = 2 > 1, this conic is a hyperbola with c = 3.

16. Copyright © 2007 Pearson Education, Inc. Slide 6-16 6.3 Finding Equations of Conics using Eccentricity The focus is on the x-axis, so the x 2 -term is positive. The equation is

17. Copyright © 2007 Pearson Education, Inc. Slide 6-17 6.3 Finding Equations of Conics using Eccentricity (b)Since the conic is an ellipse. The vertex at (0, –8) indicates that the vertices lie on the y-axis and a = 8. Since the equation is

18. Copyright © 2007 Pearson Education, Inc. Slide 6-18 6.3 Applying an Ellipse to the Orbit of a Planet Example The orbit of Mars is an ellipse with the sun at one focus. The eccentricity is.0935, and the closest distance that Mars comes to the sun is 128.5 million miles. Find the maximum distance of Mars from the sun. Solution Using the given figure, Mars is - closest to the sun on the right -farthest from the sun on the left Therefore, -smallest distance is a – c -greatest distance is a + c

19. Copyright © 2007 Pearson Education, Inc. Slide 6-19 6.3 Applying an Ellipse to the Orbit of a Planet Since a – c = 128.5, c = a – 128.5. Using e =.0935, we find a. The maximum distance of Mars from the sun is about 155.1 million miles.