1008 CHAPTER 10 Analytic Geometry For an ellipse, eccentricity is a measure of its “roundness.” The constant ratio in the definition is e = c a , where (as before) c is the distance from the center of the figure to a focus, and a is the distance from the center to a vertex. By the definition of an ellipse, a2 7b2 and c = 2a2 - b2. Thus, for the ellipse, we have the following. 0 6c 6a Inequality fact for an ellipse 0 6 c a 61 Divide by a. 0 6e 61 e = c a Thus, every ellipse has eccentricity between 0 and 1. If a is constant, letting c approach 0 would force the ratio c a to approach 0, which also forces b to approach a (so that 2a2 - b2 = c would approach 0). Because b determines the endpoints of the minor axis, this means that the lengths of the major and minor axes are almost the same, producing an ellipse very close in shape to a circle when e is very close to 0. In a similar manner, if e approaches 1, then b will approach 0. The path of Earth around the sun is an ellipse that is very nearly circular. In fact, for this ellipse, e ≈0.017. On the other hand, the path of Halley’s comet is a very flat ellipse, with e ≈0.97. Figure 24 compares ellipses with different eccentricities. The locations of the foci are shown in each case. x y e = 1 4 x y e = 1 2 x y e = 3 4 Figure 24 Eccentricities of ellipses EXAMPLE 6 Finding Eccentricity from Equations of Ellipses Find the eccentricity e of each ellipse. Round answers to the nearest hundredth. (a) x2 9 + y2 16 = 1 (b) 5x2 + 10y2 = 50 SOLUTION (a) Because 16 79, a2 = 16 and thus a = 4. To find c, use c2 = a2 - b2. c2 = a2 - b2 Relationship for ellipses c2 = 16 - 9 Let a2 = 16 and b2 = 9. c = 27 Subtract. Take the positive square root. The equation of a circle with center 1h, k2 and radius r can be written as follows. 1x - h22 + 1y - k22 = r2 Center-radius form of a circle 1x - h22 r 2 + 1y - k22 r 2 = 1 Divide by r2. In a circle, the foci coincide with the center, so a = b, c = 2a2 - b2 = 0, and e = 0. Thus, every circle has eccentricity 0.
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