698 CHAPTER 10 Analytic Geometry Finding an Equation of an Ellipse, Center Not at the Origin Find an equation of the ellipse with center at 2, 3 , ( ) − one focus at 3, 3 , ( ) − and one vertex at 5, 3 . ( ) − Graph the equation. EXAMPLE 6 Figure 29 x y 2 9 3 8 1 2 2 ( ) ( ) − + − = x y 6 22 2 (2, 23 1 2 ) 2 (2, 23 2 2 ) (2, 23) (3, 23) (1, 23) V2 5 (5, 23) V1 5 (21, 23) 2 F1 F2 Figure 28 x (h 1 a, k) (h , k) (h 2 a, k) x (h, k 1 a) (h , k) (h, k 2 a) (a) (b) Major axis (h 1 c, k) (h 2 c, k) Major axis (h, k 1 c) (h, k 2 c) y y (x 2 h)2 –––––– (y 2 k)2 –––––– b2 1 5 1 (x 2 h)2 –––––– b2 (y 2 k)2 –––––– 1 5 1 a2 a2 Solution The center is at h k , 2, 3, ( ) ( ) = − so h 2 = and k 3. = − Note that the center, focus, and vertex all lie on the line y 3. = − Therefore, the major axis is parallel to the x-axis. The distance from the center 2, 3 ( ) − to a focus 3, 3 ( ) − is c 1; = the distance from the center 2, 3 ( ) − to a vertex 5, 3 ( ) − is a 3. = Then b a c 9 1 8. 2 2 2 = − = − = The form of the equation is x h a y k b x y 1 2 9 3 8 1 2 2 2 2 2 2 ( ) ( ) ( ) ( ) − + − = − + + = = =− = = h k a b 2, 3, 9, 8 2 2 The major axis is parallel to the x-axis, so the vertices are a 3 = units left and right of the center 2, 3 . ( ) − Therefore, the vertices are ( ) ( ) ( ) ( ) = − − = − − = + − = − V V 2 3, 3 1, 3 and 2 3, 3 5, 3 1 2 Note that the vertex 5, 3 ( ) − agrees with the information given in the problem. Since c 1 = and the major axis is parallel to the x-axis, the foci are 1 unit left and right of the center. Therefore, the foci are F F 2 1, 3 1, 3 and 2 1, 3 3, 3 1 2 ( ) ( ) ( ) ( ) = − − = − = + − = − Finally, use the value of b 2 2 = to find the two points above and below the center. ( ) ( ) − − − + 2, 3 22 and 2, 3 22 Use these two points and the vertices to obtain the graph. See Figure 29. Now Work PROBLEM 57 Using a Graphing Utility to Graph an Ellipse, Center Not at the Origin Using a graphing utility, graph the ellipse: x y 2 9 3 8 1 2 2 ( ) ( ) − + + = EXAMPLE 7
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