SECTION 10.3 The Ellipse 699 To graph the equation using a TI-84 Plus CE graphing calculator, first solve the equation x y 2 9 3 8 1 2 2 ( ) ( ) − + + = for y. y x y x y x y x 3 8 1 2 9 3 8 1 2 9 3 8 1 2 9 3 8 1 2 9 2 2 2 2 2 2 ( ) ( ) ( ) ( ) ( ) ( ) + = − − + = − ⎡ − ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ + = ± − ⎡ − ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ = − ± − ⎡ − ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ Figure 30(a) shows the graphs of Y x 3 8 1 2 9 1 2 ( ) = − + − ⎡ − ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ and Y x 3 8 1 2 9 2 2 ( ) = − − − ⎡ − ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ using a TI-84 Plus CE. Figure 30(b) shows the graph obtained using Desmos. 1 26 23.6 7.6 (x 2 2)2 9 Y1 5 23 1 1 2 8( ) (x 2 2)2 9 Y2 5 23 2 1 2 8( ) (a) Figure 30 (b) Solution Subtract ( ) −x 2 9 2 from each side. Multiply each side by 8. Use the Square Root Method. Subtract 3 from each side. Analyzing the Equation of an Ellipse Analyze the equation x y x y 4 8 4 4 0. 2 2 + − + + = EXAMPLE 8 Figure 31 x y x y 4 8 4 4 0 2 2 + − + + = x y 24 11, 22 2 3 2 11, 22 1 3 2 (1, 24) (0, 22) (2, 22) (1, 22) (1, 0) (a) (b) Solution Complete the squares in x and in y. ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) + − + + = − + + = − − + + = − − + + + + = − + + − + + = − + + = x y x y x x y y x x y y x x y y x y x y 4 8 4 4 0 4 8 4 4 4 2 4 4 4 2 1 4 4 4 4 4 4 1 2 4 1 2 4 1 2 2 2 2 2 2 2 2 2 2 2 2 This is the equation of an ellipse with center at 1, 2 ( ) − and major axis parallel to the y-axis. Since a 4 2 = and b 1, 2 = we have c a b 4 1 3. 2 2 2 = − = − = The vertices are at h k a , 1, 2 2, ( ) ( ) ± = − ± or 1, 4 ( ) − and 1,0 . ( ) The foci are at h k c , 1, 2 3, ( ) ( ) ± = − ± or 1, 2 3 ( ) − − and 1, 2 3. ( ) − + Figure 31(a) shows the graph drawn by hand. Figure 31(b) shows the graph obtained using GeoGebra. Group like variables; place the constant on the right side. Factor out 4 from the first two terms. Complete each square. Factor. Divide both sides by 4. Now Work PROBLEM 49
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