Polar Equations of Conics

1. Precalculus Lesson 8.5 Polar Equations of Conics

2. Quick Review

3. 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.

4. 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.)

5. Focus-Directrix Eccentricity Relationship If Pis a point of a conic section, F is the conic’s focus, and D is the point of the directrix closest to P, then where e is a constant and the eccentricity of the conic. Moreover, the conic is  a hyperbola if e > 1,  a parabola if e = 1,  an ellipse if e < 1.

6. The Geometric Structure of a Conic Section

7. A Conic Section in the Polar Plane

8. Three Types of Conics for r = ke/(1+ecosθ) Directrix Directrix Directrix P D P D P D F(0,0) F(0,0) F(0,0) x = k x = k x = k Parabola Ellipse Hyperbola

9. Polar Equations for Conics Two standard orientations of a conic in the polar plane are as follows. Focus at pole Focus at pole Directrixx = k Directrixx = k

10. Polar Equations for Conics The other two standard orientations of a conic in the polar plane are as follows. Directrixy = k Focus at pole Focus at pole Directrixy = k

11. Example Writing Polar Equations of Conics

12. Example Identifying Conics from Their Polar Equations Note, the sign in the denominator dictates the sign of the directrix.

13. Example Writing a Conic Section in Polar Form

14. Example Writing a Conic Section in Polar Form

15. Example Writing a Conic Section in Polar Form

16. Homework: Text pg683 Exercises # 4-40 (intervals of 4)