SECTION 10.3 The Ellipse 695 Figure 23 3 23 24.8 4.8 x2 16 Y1 5 7 1 2 ( ) x2 16 Y2 5 2 7 1 2 ( ) First, solve x y 16 7 1 2 2 + = for y. ( ) ( ) = − = − = ± − y x y x y x 7 1 16 7 1 16 7 1 16 2 2 2 2 2 Figure 23* shows the graphs of Y x 7 1 16 1 2 ( ) = − and Y x 7 1 16 2 2 ( ) = − − on a TI-84 Plus CE. Graphing an Ellipse Using a Graphing Utility Use a graphing utility to graph the ellipse x y 16 7 1. 2 2 + = EXAMPLE 2 Solution Subtract x 16 2 from each side. Multiply both sides by 7. Use the Square Root Method. In Figure 23 a square screen is used.As with circles and parabolas, this is done to avoid a distorted view of the graph. An equation of the form of equation (2), with a b, > is the equation of an ellipse with center at the origin, foci on the x-axis at c, 0 ( ) − and c, 0 , ( ) where = − c a b , 2 2 2 and major axis along the x-axis. For the remainder of this section, the direction “Analyze the equation” means to find the center, major axis, foci, and vertices of the ellipse and graph it. Figure 24 x y 25 9 1 2 2 + = x y 26 6 6 V2 5 (5, 0) F 2 5 (4, 0) F1 5 (24, 0) V1 5 (25, 0) (0, 3) (0, 23) (a) (b) Analyzing the Equation of an Ellipse Analyze the equation x y 25 9 1. 2 2 + = EXAMPLE 3 Solution The equation is of the form of equation (2), with a 25 2 = and b 9. 2 = The equation is that of an ellipse with center at 0, 0 ( ) and major axis along the x-axis. The vertices are at a, 0 5, 0 . ( ) ( ) ± = ± Because b a c , 2 2 2 = − this means c a b 25 9 16 2 2 2 = − = − = The foci are at c, 0 4, 0 . ( ) ( ) ± = ± The y-intercepts are b 0, 0, 3. ( ) ( ) ± = ± Figure 24(a) shows the graph drawn by hand. Figure 24(b) shows the graph obtained using Desmos.Again, notice that Desmos does not require that we solve the equation for y. Now Work PROBLEM 19 *The initial viewing window selected was X X Y Y min 4, max 4, min 3, max 3. = − = = − = Then we used the ZOOM-SQUARE option to obtain the window shown.
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