696 CHAPTER 10 Analytic Geometry If the major axis of an ellipse with center at 0, 0 ( ) lies on the y -axis, the foci are at c 0, ( ) − and c 0, . ( ) Using the same steps as before, the definition of an ellipse leads to the following result. THEOREM Equation of an Ellipse: Center at ( ) 0, 0 ; Major Axis along the y -Axis An equation of the ellipse with center at 0, 0 , ( ) foci at c 0, ( ) − and c 0, , ( ) and vertices at a 0, ( ) − and a 0, ( ) is x b y a a b b a c 1 where 0 and 2 2 2 2 2 2 2 + = > > = − (3) The major axis is the y -axis. Figure 25 x b y a a b 1, 0 2 2 2 2 + = > > x b a c y (b, 0) (2b, 0) V2 5 (0, a) V1 5 (0, 2a) F2 5 (0, c) F1 5 (0, 2c) Figure 25 illustrates the graph of such an ellipse. Again, notice the right triangle formed by the points at b 0,0, ,0, ( ) ( ) and c 0, , ( ) so that ( ) = + = − a b c b a c or . 2 2 2 2 2 2 Look closely at equations (2) and (3). Although they may look alike, there is a difference! In equation (2), the larger number, a ,2 is in the denominator of the x -term, 2 so the major axis of the ellipse is along the x -axis. In equation (3), the larger number, a ,2 is in the denominator of the y -term, 2 so the major axis is along the y -axis. Figure 26 x y 9 9 2 2 + = x y 23 3 3 23 2 5 (0, 3) V1 5 (0, 23) (1, 0) (21, 0) (a) 3.75 23.75 26 6 (b) V F2 5 (0, 2 2) F1 5 (0, 22 2) Y1 5 3 1 2 x2 Y2 5 23 1 2 x2 Analyzing the Equation of an Ellipse Analyze the equation x y 9 9. 2 2 + = EXAMPLE 4 Solution To put the equation in proper form, divide both sides by 9. x y 9 1 2 2 + = The larger denominator, 9, is in the y -term 2 so, based on equation (3), this is the equation of an ellipse with center at the origin and major axis along the y -axis. Also, a b 9, 1, 2 2 = = and c a b 9 1 8. 2 2 2 = − = − = The vertices are at a 0, 0, 3, ( ) ( ) ± = ± and the foci are at c 0, 0, 22. ( ) ( ) ± = ± The x -intercepts are at ( ) ( ) ± = ± b, 0 1, 0 . Figure 26(a) shows the graph drawn by hand. Figure 26(b) shows the graph using a TI-84 Plus CE. Now Work PROBLEM 23
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