714 CHAPTER 10 Analytic Geometry Analyzing the Equation of a Hyperbola Analyze the equation x y x y 4 2 16 11 0. 2 2 − + − − + = Solution EXAMPLE 9 Figure 48 − + − − + = x y x y 4 2 16 11 0 2 2 Transverse V2 5 (21, 3) (b) (a) x y 5 axis 5 2 V1 5 (21, 1) (1, 2) (23, 2) 5 F2 5 (21, 2 1 5 ) F1 5 (21, 2 2 5 ) Complete the squares in x and in y. x y x y x x y y x x y y x y y x 4 2 16 11 0 2 4 4 11 2 1 4 4 4 11 1 16 1 4 2 4 2 1 4 1 2 2 2 2 2 2 2 2 2 2 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) − + − − + = − + + − = − −+++ −+=−−+ − + + − = − − + = This is the equation of a hyperbola with center at 1, 2 ( ) − and transverse axis parallel to the y-axis. Also, a 1 2 = and b 4, 2 = so c a b 5. 2 2 2 = + = Since the transverse axis is parallel to the y-axis, the vertices and foci are located a and c units above and below the center,respectively.The vertices are at h k a , 1, 2 1 , ( ) ( ) ± = − ± or 1, 1 ( ) − and 1,3 . ( ) − The foci are at h k c , 1, 2 5 . ( ) ( ) ± = − ± The asymptotes are y x 2 1 2 1 ( ) − = + and y x 2 1 2 1 . ( ) − = − + Figure 48(a) shows the graph drawn by hand. Figure 48(b) shows the graph obtained using GeoGebra. Group terms. Complete each square. Factor. Divide both sides by 4. Now Work PROBLEM 57 4 Solve Applied Problems Involving Hyperbolas Look at Figure 49. Suppose that three microphones are located at points O O, , 1 2 and O3 (the foci of the two hyperbolas). In addition, suppose that a gun is fired at S and the microphone at O1 records the gunshot 1 second after the microphone at O .2 Because sound travels at about 1100 feet per second, we conclude that the microphone at O1 is 1100 feet farther from the gunshot than O .2 This situation is modeled by placing S on a branch of a hyperbola with foci at O1 and O .2 (Do you see why? The difference of the distances from S to O1 and from S to O2 is the constant 1100.) If the third microphone at O3 records the gunshot 2 seconds after O,1 then S lies on a branch of a second hyperbola with foci at O1 and O .3 In this case, the constant difference will be 2200. The intersection of the two hyperbolas identifies the location of S. Figure 49 O3 O1 O2 S
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