SECTION 10.4 The Hyperbola 711 The asymptotes of a hyperbola are not part of the hyperbola, but they serve as a guide for graphing the hyperbola. For example, suppose that we want to graph the equation x a y b 1 2 2 2 2 − = Begin by plotting the vertices a, 0 ( ) − and a, 0 . ( ) Then plot the points b 0, ( ) − and b 0, ( ) and use these four points to construct a rectangle, as shown in Figure 43. The diagonals of this rectangle have slopes b a and b a , − and their extensions are the asymptotes of the hyperbola, y b a x = and y b a x. = − If we graph the asymptotes, we can use them to establish the “opening” of the hyperbola and avoid plotting other points. THEOREM Asymptotes of a Hyperbola; Transverse Axis along the y -Axis The hyperbola y a x b 1 2 2 2 2 − = has the two oblique asymptotes y a b x y a b x and = = − (5) You are asked to prove this result in Problem 88. For the remainder of this section, the direction “ Analyze the equation ” means to find the center, transverse axis, vertices, foci, and asymptotes of the hyperbola and graph it. Figure 43 − = x a y b 1 2 2 2 2 x y (0, b) (0, 2b) (2a, 0) (a, 0) y 5 xb – a y 5 2 xb – a Analyzing the Equation of a Hyperbola Analyze the equation y x 4 1. 2 2 − = EXAMPLE 6 Figure 44 − = y x 4 1 2 2 x (0, 22) (0, 2) y 5 22x y 5 2x y 25 (21, 0) (1, 0) 25 5 5 10, 52 10, 2 52 (a) (b) Solution Since the x -term 2 is subtracted from the y -term, 2 the equation is of the form of equation (3) and is a hyperbola with center at the origin and transverse axis along the y -axis. Comparing the equation to equation (3), note that a b 4, 1, 2 2 = = and c a b 5. 2 2 2 = + = The vertices are at a 0, 0, 2, ( ) ( ) ± = ± and the foci are at c 0, 0, 5 . ( ) ( ) ± = ± Using equation (5) with a 2 = and b 1, = the asymptotes are the lines y a b x x2 = = and y a b x x2 . = − = − Form the rectangle containing the points a 0, 0, 2 ( ) ( ) ± = ± and b, 0 1, 0 . ( ) ( ) ± = ± The extensions of the diagonals of the rectangle are the asymptotes. Now graph the asymptotes and the hyperbola. See Figure 44(a) for the graph drawn by hand. Figure 44(b) shows the graph obtained using Desmos.
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