SECTION 10.6 Polar Equations of Conics 729 DEFINITION Conic Let D denote a fixed line called the directrix ; let F denote a fixed point called the focus , which is not on D; and let e be a fixed positive number called the eccentricity . A conic is the set of points P in a plane for which the ratio of the distance from F to P to the distance from D to P equals e. That is, a conic is the collection of points P for which d F P d D P e , , ( ) ( ) = (1) • If e 1, = the conic is a parabola . • If e 1, < the conic is an ellipse . • If e 1, > the conic is a hyperbola . Figure 57 u p d(D, P) P 5 (r, u) Q Polar axis r Pole O (Focus F) Directrix D Observe that if e 1, = the definition of a parabola in equation (1) is exactly the same as the definition used earlier in Section 10.2. In the case of an ellipse, the major axis is a line through the focus perpendicular to the directrix. In the case of a hyperbola, the transverse axis is a line through the focus perpendicular to the directrix. For both an ellipse and a hyperbola, the eccentricity e satisfies e c a = (2) where c is the distance from the center to the focus, and a is the distance from the center to a vertex. Just as we did earlier using rectangular coordinates, we derive equations for the conics in polar coordinates by choosing a convenient position for the focus F and the directrix D. The focus F is positioned at the pole, and the directrix D is either parallel or perpendicular to the polar axis. Suppose that we start with the directrix D perpendicular to the polar axis at a distance p units to the left of the pole (the focus F ). See Figure 57. If P r, θ ( ) = is any point on the conic, then, by equation (1), d F P d D P e d F P e d D P , , or , , ( ) ( ) ( ) ( ) = = ⋅ (3) Now use the point Q obtained by dropping the perpendicular from P to the polar axis to calculate d D P , . ( ) d D P p d O Q p r , , cosθ ( ) ( ) = + = + (4) Since the focus F is at the pole (origin), it follows that d F P d O P r , , ( ) ( ) = = (5) Use the results in equations (4) and (5) in equation (3). Then d F P e d D P r e p r r ep er r er ep r e ep r ep e , , cos cos cos 1 cos 1 cos θ θ θ θ θ ( ) ( ) ( ) ( ) = ⋅ = + = + − = − = = − Equation (3) d F P r d D P p r , ; , cosθ ( ) ( ) = = +
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