1. Polar Equations of Conics
4/26/2017 7:34 AM
Precalculus
Lesson
8.5
Polar Equations of Conics
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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 P is 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 D P
F(0,0)
x = k
Ellipse
Directrix D P
F(0,0)
x = k
Hyperbola
Directrix D P
F(0,0)
x = k
Parabola

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

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

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)