1029 10.4 Summary of the Conic Sections Because of the -36, we might think that this equation does not have a graph. However, dividing each side by -36 reveals that the graph is that of a hyperbola.1x - 122 4 - 1y - 222 9 = 1 Divide by -36, and rearrange terms. This hyperbola has center 11, 22. The graph is shown in Figure 40. x 4y2 – 16y – 9x2 + 18x = –43 y (–1, 2) (3, 2) –2 0 5 –1 3 Figure 40 S Now Try Exercise 43. Be careful here! x = 8 2 2 4 x y P 0 Figure 41 x y x = 1 P 3 4 0 Figure 42 Geometric Definition of Conic Sections A parabola was defined as the set of points in a plane equidistant from a fixed point (focus) and a fixed line (directrix). A parabola has eccentricity 1. This definition can be generalized to apply to ellipses and hyperbolas. Figure 41 shows an ellipse with a = 4, c = 2, and e = 1 2 . The line x = 8 is shown also. For any point P on the ellipse, distance of P from the focus = 1 2 3distance of P from the line4. Figure 42 shows a hyperbola with a = 2, c = 4, and e = 2, along with the line x = 1. For any point P on the hyperbola, distance of P from the focus = 23distance of P from the line4. The following geometric definition applies to all conic sections except circles, which have e = 0. Geometric Definition of a Conic Section Given a fixed point F (focus), a fixed line L (directrix), and a positive number e, the set of all points P in the plane such that distance of P from F =e # 3 distance of P from L4 is a conic section of eccentricity e. The conic section is a parabola when e =1, an ellipse when 0 *e *1, and a hyperbola when e +1.
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