SECTION 10.3 The Ellipse 697 Finding an Equation of an Ellipse Find an equation of the ellipse having one focus at 0, 2 ( ) and vertices at 0, 3 ( ) − and 0, 3 . ( ) Graph the equation. EXAMPLE 5 (b) Solution Plot the given focus and vertices, and note that the major axis is the y-axis. Because the vertices are at 0, 3 ( ) − and 0,3 , ( ) the center of the ellipse is at their midpoint, the origin. The distance from the center, 0, 0 , ( ) to the given focus, 0, 2 , ( ) is c 2. = The distance from the center, 0, 0 , ( ) to one of the vertices, 0, 3 , ( ) is a 3. = So b a c 9 4 5. 2 2 2 = − = − = The form of the equation of the ellipse is given by equation (3). x b y a x y 1 5 9 1 2 2 2 2 2 2 + = + = Figure 27(a) shows the graph drawn by hand. Figure 27(b) shows the graph obtained using GeoGebra. Now Work PROBLEM 31 A circle may be considered a special kind of ellipse. To see why, let a b = in equation (2) or (3). Then x a y a x y a 1 2 2 2 2 2 2 2 + = + = This is the equation of a circle with center at the origin and radius a. The value of c is c a b 0 2 2 2 = − = ↑ =a b This indicates that the closer the two foci of an ellipse are to the center, the more the ellipse will look like a circle. 2 Analyze Ellipses with Center at ( ) h k , If an ellipse with center at the origin and major axis coinciding with a coordinate axis is shifted horizontally h units and then vertically k units, the result is an ellipse with center at h k , ( ) and major axis parallel to a coordinate axis. The equations of such ellipses have the same forms as those given in equations (2) and (3), except that x is replaced by x h − (the horizontal shift) and y is replaced by y k − (the vertical shift). Table 3 gives the forms of the equations of such ellipses, and Figure 28 on the next page shows their graphs. TIP Rather than memorizing Table 3, use transformations (shift horizontally h units, vertically k units), along with the facts that a represents the distance from the center to the vertices, c represents the distance from the center to the foci, and ( ) = − = − b a c c a b . or 2 2 2 2 2 2 j Equations of an Ellipse: Center at ( ) h k , ; Major Axis Parallel to a Coordinate Axis Center Major Axis Foci Vertices Equation h k , ( ) Parallel to the x-axis h c k , ( ) + ( ) + a h k , ( ) ( ) − + − = a x h y k b 1 2 2 2 2 h c k , ( ) − ( ) −a h k , > > = − a a b b c 0 and 2 2 2 h k , ( ) Parallel to the y-axis h k c , ( ) + ( ) + a h k , ( ) ( ) − + − = a x h b y k 1 2 2 2 2 h k c , ( ) − ( ) −a h k , > > = − a a b b c 0 and 2 2 2 Table 3 Figure 27 x y 5 9 1 2 2 + = x y 12 , 02 23 3 3 23 V2 5 (0, 3) V1 5 (0, 23) 5 1 , 02 5 F2 5 (0, 2) F1 5 (0, 22) (a)
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