1006 CHAPTER 10 Analytic Geometry Standard Forms for Ellipses Centered at 1h, k2 An ellipse with center at 1h, k2 and either a horizontal or a vertical major axis of length 2a satisfies one of the following equations, where a 7b 70 and c2 = a2 - b2 with c 70. 1x −h22 a2 + 1 y −k22 b2 =1 Major axis: horizontal; vertices: 1h{a, k2; foci: 1h{c, k2 1x −h22 b2 + 1 y −k22 a2 =1 Major axis: vertical; vertices: 1h, k {a2; foci: 1h, k {c2 F9 F x y 0 a a c b c b (h, k) Vertical major axis F9 F x y 0 c c b (h, k) b a a Horizontal major axis When graphing ellipses, remember that the location of a2 (the greater denominator) determines whether the ellipse has a horizontal or a vertical major axis. EXAMPLE 4 Graphing an Ellipse Translated Away from the Origin Graph 1x - 222 9 + 1y + 122 16 = 1. Give the foci, domain, and range. SOLUTION The graph of this equation is an ellipse centered at 12, -12. Because a 7b for ellipses, a = 4 and b = 3. This ellipse has a vertical major axis because a2 = 16 is associated with y2. The vertices are located a distance of a = 4 units directly above and below the center, at 12, 32 and 12, -52. Two other points on the ellipse, located a distance of b = 3 units to the left and right of the center, are 1-1, -12 and 15, -12. The foci are found using the following equation. c2 = a2 - b2 Relationship for ellipses c2 = 16 - 9 Let a2 = 16 and b2 = 9. c2 = 7 Subtract. c = 27 Take the positive square root because c 70. The foci are located on the major axis a distance of c = 27 (approximately 2.6) units above and below the center 12, -12, at A2, -1 + 27 B and A2, -1 - 27 B. See the graph in Figure 21. The domain is 3-1, 54, and the range is 3-5, 34. S Now Try Exercise 17. y 5 –1 –5 3 x + = 1 (y + 1)2 16 (x – 2)2 9 (5, –1) (–1, –1) F9(2, –1–Ë7) V9(2, –5) V(2, 3) F(2, –1+Ë7) (2, –1) Figure 21
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