712 CHAPTER 10 Analytic Geometry Analyzing the Equation of a Hyperbola Analyze the equation x y 9 4 36. 2 2 − = Solution EXAMPLE 7 Divide both sides of the equation by 36 to put the equation in proper form. x y 4 9 1 2 2 − = The center of the hyperbola is the origin. Since the y -term 2 is subtracted from the x -term, 2 the transverse axis is along the x-axis, and the vertices and foci will lie on the x-axis. Using equation (2), note that a b 4, 9, 2 2 = = and c a b 13. 2 2 2 = + = The vertices are a 2 = units left and right of the center at a, 0 2, 0 , ( ) ( ) ± = ± the foci are c 13 = units left and right of the center at c, 0 13, 0 , ( ) ( ) ± = ± and the asymptotes have the equations y b a x x y b a x x 3 2 and 3 2 = = = − = − To graph the hyperbola, form the rectangle containing the points a, 0 ( ) ± and b 0, , ( ) ± thatis, 2,0 , 2,0 , 0, 3 , ( ) ( ) ( ) − − and 0,3 . ( ) The extensions of the diagonals of the rectangle are the asymptotes. See Figure 45(a) for the graph drawn by hand. Figure 45(b) shows the graph obtained using a TI-84 Plus CE. Figure 45 − = x y 9 4 36 2 2 x y 5 5 5 (0, 3) (0, 23) 25 V1 5 (22, 0) V (2, 0) (a) 2 2 5 y 5 x3 – 2 5 2 3 – 2 y x F2 F1 x2 4 3 2 1 5 Y1 5 28 8 (b) x 4 Y2 5 23 2 1 25 2 3 2 Y3 5 2 x 3 2 Y4 5 x Now Work PROBLEM 33 3 Analyze Hyperbolas with Center at ( ) h k , If a hyperbola with center at the origin and transverse axis coinciding with a coordinate axis is shifted horizontally h units and then vertically k units, the result is a hyperbola with center at h k , ( ) and transverse axis parallel to a coordinate axis. The equations of such hyperbolas 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 4 gives the forms of the equations of such hyperbolas. See Figure 46 on the next page for typical graphs. Equations of a Hyperbola: Center at ( ) h, k ; Transverse Axis Parallel to a Coordinate Axis Center Transverse Axis Foci Vertices Equation Asymptotes ( ) h k , Parallel to the x-axis ( ) ± h c k , ( ) ± a h k , a a ( ) ( ) − − − = = − x h y k b b c 1, 2 2 2 2 2 2 2 a ( ) − = ± − y k b x h ( ) h k , Parallel to the y-axis ( ) ± h k c , a ( ) ± h k , a a ( ) ( ) − − − = = − y k x h b b c 1, 2 2 2 2 2 2 2 a ( ) − = ± − y k b x h Table 4 Seeing the Concept Refer to Figure 45(b). Create a TABLE using Y1 and Y4 with = x 10, 100, 1000, and 10,000. Compare the values of Y1 and Y .4 Repeat for Y2 and Y .3 Now, create a TABLE using Y1 and Y3 with =− − − x 10, 100, 1000, and −10,000. Repeat for Y2 and Y .4
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