Conic Sections in Polar Coordinates Lesson 10.6

Definition of Parabola Set of points equal distance from a point and a line Point is the focus Line is the directrix If the ratio of the two distances is different from 1, other curves result •

General Definition of a Conic Section Given a fixed line L and a fixed point F A conic section is the set of all points P in the plane such that F • d(P, F) • d(P, L) L Note: This e stands for eccentricity. It is not the same as e = 2.71828

General Definition of a Conic Section When e has different values, different curves result 0 < e < 1 The conic is an ellipse e = 1 The conic is a parabola e > 1 The conic is a hyperbola Note: The distances are positive e is always greater than zero

Polar equations of Conic Sections A polar equation that has one of the following forms is a conic section When cos is used, major axis horizontal Directrix at x = p When sin is used, major axis vertical Directrix at y = p

Note the false asymptotes Example Given Identify the conic What is the eccentricity? e = ______ Graph the conic Note the false asymptotes

Special Situation Consider Now graph Note it is rotated by -π/6 Eccentricity = ? Conic = ? Now graph Note it is rotated by -π/6

Finding the Polar Equation Given directrix y = -5 and e = 1 What is the conic? Which equation to use? 5 • 1 y = - 5

Assignment Lesson 10.6 Page 438 Exercises 1 – 19 odd