1. Adapted by JMerrill, 2010

2. Copyright © by Houghton Mifflin Company, Inc. All rights reserved.2 Definition: Conic The locus of a point in the plane which moves so that its distance from a fixed point (focus) is in a constant ratio to its distance from a fixed line (directrix) is a conic. The constant ratio is the eccentricity (or measure of the flatness) of the conic and is denoted by e. Alternative Definition of a Conic: 0 P F = (0, 0) Directri x Q Locus

3. Copyright © by Houghton Mifflin Company, Inc. All rights reserved.3 Ellipse P 0 F = (0, 0) Directri x Q

4. Copyright © by Houghton Mifflin Company, Inc. All rights reserved.4 Parabola P 0 F = (0, 0) Directri x Q Parabola

5. Copyright © by Houghton Mifflin Company, Inc. All rights reserved.5 Hyperbola P 0 F = (0, 0) Directri x Q Hyperbola

6. Copyright © by Houghton Mifflin Company, Inc. All rights reserved.6 Definition: Polar Equations of Conics The graph of a polar equation of the form Polar Equations of Conics: is a conic, where e > 0 is the eccentricity and |p| is the distance between the focus (pole) and the directrix. or Vertical directrix Horizontal directrix 0 P F = (0, 0) Directri x Q | p|

7. Copyright © by Houghton Mifflin Company, Inc. All rights reserved.7 Example: Identifying Conics Example: Identify the type of conic represented by the equation The graph is an ellipse with a distance of 12 between the pole and the directrix. Divide the numerator and denominator by 2 to rewrite in the form

8. Copyright © by Houghton Mifflin Company, Inc. All rights reserved.8 Graphing Utility: Graphing a Polar Equation Graphing Utility: Graph the conic given by the equation Mode Menu: Set to polar mode. –6 6 6 The graph is an ellipse with a distance of 12 between the pole and the directrix.

9. Copyright © by Houghton Mifflin Company, Inc. All rights reserved.9 Finding Equations Given the Directrix If p is the distance between the directrix and the pole, then one of the following four formulas will be used to find the polar equation for the conic 1. Horizontal directrix above the pole: 2. Horizontal directrix below the pole: 3. Vertical directrix to the right of the pole: 4. Vertical directrix to the left of the pole:

10. Copyright © by Houghton Mifflin Company, Inc. All rights reserved.10 Example: Analyzing a Graph of a polar Eqauation Example: Analyze the graph of the polar equation Divide the numerator and denominator by 3. Example continues.

11. Copyright © by Houghton Mifflin Company, Inc. All rights reserved.11 Example: Analyzing a Graph of a polar Eqauation Example continued: 24 – 18 – 4 24 Use trace to find the vertices

12. Copyright © by Houghton Mifflin Company, Inc. All rights reserved.12 Example: Finding Polar Equations Example: Find the polar equation of the parabola whose focus is the pole and whose directrix is y = – 4. The directrix is horizontal and below the pole. x -4 4 y 4 8 e = 1 parabola