1017 10.3 Hyperbolas Standard Forms of Equations for Hyperbolas The hyperbola with center at the origin and equation x2 a2 − y2 b2 =1 has vertices 1{a, 02, asymptotes y = { b a x, and foci 1{c, 02, where c2 = a2 + b2. The hyperbola with center at the origin and equation y2 a2 − x2 b2 =1 has vertices 10, {a2, asymptotes y = { a b x, and foci 10, {c2, where c2 = a2 + b2. x y –c –a a c b –b Transverse axis on x-axis x y –a –b b a c –c Transverse axis on y-axis EXAMPLE 2 Graphing a Hyperbola Graph 25y2 - 4x2 = 100. Give the equations of the asymptotes, and the foci, domain, and range. SOLUTION y2 4 - x2 25 = 1 Divide by 100, and write in standard form. This hyperbola is centered at the origin, has foci on the y-axis, and has vertices 10, 22 and 10, -22. The equations of the asymptotes are found as follows. y = { a b x Asymptotes for a hyperbola with vertical transverse axis y = { 2 5 x Let a = 2 and b = 5. To graph the asymptotes, use the points 15, 22, 15, -22, 1-5, 22, and 1-5, -22 to determine the fundamental rectangle. The extended diagonals of this rectangle are the asymptotes for the graph, as shown in Figure 32. The foci are located on the y-axis, c units above and below the origin. c2 = a2 + b2 Relationship for hyperbolas c2 = 4 + 25 Let a2 = 4 and b2 = 25. c2 = 29 Add. c = 229 Take the positive square root because c 70. The coordinates of the foci are A0, 229 B and A0, -229 B. The domain of the relation is 1-∞, ∞2, and the range is 1-∞, -24 ´32, ∞2. S Now Try Exercise 17. x y –5 5 2 –2 – = 1 4 y2 25 x2 25y2 – 4x2 = 100 F9(0, –Ë29) F(0, Ë29) V9(0, –2) V(0, 2) Figure 32
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