9.4 THE HYPERBOLA

A hyperbola is the collection of points in the plane the difference of whose distances from two fixed points, called the foci, is a constant.

Theorem. Equation of a Hyberbola; Center at Theorem Equation of a Hyberbola; Center at (0, 0); Foci at ( + c, 0); Vertices at ( + a, 0); Transverse Axis along the x-Axis An equation of the hyperbola with center at (0, 0), foci at ( - c, 0) and (c, 0), and vertices at ( - a, 0) and (a, 0) is The transverse axis is the x-axis.

Theorem. Equation of a Hyberbola; Center at Theorem Equation of a Hyberbola; Center at (0, 0); Foci at ( 0, + c); Vertices at (0, + a); Transverse Axis along the y-Axis An equation of the hyperbola with center at (0, 0), foci at (0, - c) and (0, c), and vertices at (0, - a) and (0, a) is The transverse axis is the y-axis.

Theorem Asymptotes of a Hyperbola The hyperbola has the two oblique asymptotes

Theorem Asymptotes of a Hyperbola The hyperbola has the two oblique asymptotes

Find an equation of a hyperbola with center at the origin, one focus at (0, 5) and one vertex at (0, -3). Determine the oblique asymptotes. Graph the equation by hand and using a graphing utility. Center: (0, 0) Focus: (0, 5) = (0, c) Vertex: (0, -3) = (0, -a) Transverse axis is the y-axis, thus equation is of the form

= 25 - 9 = 16 Asymptotes:

V (0, 3) F(0, 5) (-4, 0) (4, 0) F(0, -5) V (0, -3)

Hyperbola with Transverse Axis Parallel to the x-Axis; Center at (h, k) where b2 = c2 - a2.

Hyperbola with Transverse Axis Parallel to the y-Axis; Center at (h, k) where b2 = c2 - a2.

Find the center, transverse axis, vertices, foci, and asymptotes of

Center: (h, k) = (-2, 4) Transverse axis parallel to x-axis. Vertices: (h + a, k) = (-2 + 2, 4) or (-4, 4) and (0, 4)

(h, k) = (-2, 4) Asymptotes:

y - 4 = 2(x + 2) y - 4 = -2(x + 2) (-2, 8) V (-4, 4) V (0, 4) F (-6.47, 4) F (2.47, 4) C(-2,4) (-2, 0)