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Conic Sections in Polar Coordinates. Lesson 10.6. If the ratio of the two distances is different from 1, other curves result. Definition of Parabola. Set of points equal distance from a point and a line Point is the focus Line is the directrix. •. d( P, F ). •. d( P, L ).
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Conic Sections in Polar Coordinates Lesson 10.6
If the ratio of the two distances is different from 1, other curves result Definition of Parabola • Set of points equal distancefrom a point and a line • Point is the focus • Line is the directrix •
d(P, F) • d(P, L) 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 L Note: This estands for eccentricity. It is not the same as e = 2.71828
General Definition of a Conic Section • When e has differentvalues, different curvesresult • 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 y = p • When sin is used, major axis vertical • Directrix at x = p
Example • Given • Identify the conic • What is the eccentricity? • e = ______ • Graph the conic Note the false asymptotes
Special Situation • Consider • Eccentricity = ? • Conic = ? • Now graph • Note it is rotated by-π/6
5 • 1 y = - 5 Finding the Polar Equation • Given directrix y = -5 and e = 1 • What is the conic? • Which equation to use?
Assignment • Lesson 10.6 • Page 438 • Exercises 1 – 19 odd