SECTION 10.4 The Hyperbola 713 Now Work PROBLEM 43 TIP Rather than memorize Table 4, use transformations (shift horizontally h units, vertically k units), along with the facts that a represents the distance from the center to the vertices, c represents the distance from the center to the foci, and = − b c a . 2 2 2 j Figure 46 x y V2 V1 (h, k) Transverse axis (h, k) Transverse axis x y V2 V1 (b) 2 5 1 (y 2 k) 2 ––––––– a2 (x 2 h) 2 –––––– b2 (a) 2 5 1 (x 2 h) 2 ––––––– a2 (y 2 k) 2 –––––– b2 F2 F1 F2 F1 Figure 47 ( ) ( ) − − + = x y 1 4 2 5 1 2 2 x y 26 6 11, 22 1 52 4 26 11, 22 2 52 (1, 22) V1 5 (21, 22) V2 5 (3, 22) Transverse axis F 1 5 (22, 22) F2 5 (4, 22) (a) (b) Finding an Equation of a Hyperbola, Center Not at the Origin Find an equation for the hyperbola with center at 1, 2 , ( ) − one focus at 4, 2 , ( ) − and one vertex at 3, 2 . ( ) − Graph the equation. Solution EXAMPLE 8 The center is at h k , 1, 2, ( ) ( ) = − so h 1 = and k 2. = − Since the center, focus, and vertex all lie on the line y 2, = − the transverse axis is parallel to the x-axis. The distance from the center 1, 2 ( ) − to the focus 4, 2 ( ) − is c 3; = the distance from the center 1, 2 ( ) − to the vertex 3, 2 ( ) − is a 2. = Then b c a 9 4 5. 2 2 2 = − = − = The equation of the hyperbola is x h a y k b x y 1 1 4 2 5 1 2 2 2 2 2 2 ( ) ( ) ( ) ( ) − − − = − − + = Form the rectangle containing the vertices: 1, 2, 3, 2, ( ) ( ) − − − and the points ( ) ± h k b , : 1, 2 5, ( ) − − 1, 2 5. ( ) − + Extend the diagonals of the rectangle to obtain the asymptotes. See Figure 47(a) for the graph drawn by hand. Figure 47(b) shows the graph obtained using Desmos.
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