Hyperbolas Date: ______________

Horizontal transverse axis: 9.5 Hyperbolas x 2x 2 a2a2 y2y2 b2b2 –= 1 y x V 1 (–a, 0)V 2 (a, 0) Hyperbolas with Center (0,0) asymptotes: y = ± x b a vertices: (-a,0), (a,0) Foci: (-c,0), (c,0) (–c, 0) (c, 0) c² = a² + b²

Vertical transverse axis: 9.5 Hyperbolas y 2y 2 b2b2 x2x2 a2a2 –= 1 y x V 1 (0, –b) V 2 (0, b) asymptotes: y = ± x b a vertices: (0,b) (0,-b) Foci: (0,-c) (0,c) (0, c) (0, -c) c² = a² + b²

x 2x 2 36 y2y2 64 –= 1 x y a 2 = 36 a = ±6 b 2 = 64 b = ± 8 vertices: (-6,0) (6, 0) Graph the hyperbola and find the vertices.

y2y2 100 x2x2 64 –= 1 x y a 2 = 64 a = ±8 b 2 = 100 b = ± 10 vertices: (0, 10) (0,-10) Graph the hyperbola and find the vertices.

x 2x 2 y2y2 –= 400 x y a 2 = 16 a = ±4 b 2 = 25 b = ± 5 vertices: (-4,0) (4, 0) Graph the hyperbola and find the vertices x 2x 2 16 y2y2 25 –= 1

x y Find the foci of each hyperbola. Then draw the graph. x 2x 2 36 y2y2 4 –= 1 Horizontal axis: c² = a² + b² = c =√40 ≈ ±6.3 Vertices: (6,0), (-6,0) Foci: (-6.3,0), (6.3,0) asymptotes: y = ± x 1 3

Find the equation of the hyperbola with the given values. a = 263, c = 407 c² = a² + b² 407² = 263² + b² 165,649 = 69,169 + b² 96,480 = b² x 2x 2 69,169 y2y2 96,480 –= 1 x 2x 2 a2a2 y2y2 b2b2 –

Hyperbolas with Center (h,k) (x – h) 2 a2a2 (y – k) 2 b2b2 –= 1 y x V1V1 V2V2 (h,k) (y – k) 2 b2b2 (x – h) 2 a2a2 –= 1 y x V2V2 V1V1 (h,k)

(x – 2) 2 9 (y + 1) 2 16 –= 1 center: (2, –1) a 2 = 9 a = 3 c 2 = 25 vertices: (-1, -1) (5,-1) Write an equation of a hyperbola with the given characteristics. foci: (-3,-1) (7,-1) x y Horizontal axis Center is the midpoint of the vertices. c = 5 c² = a² + b² 25 = 9 + b² 16 = b² 3