1. 9.4
THE HYPERBOLA

2. 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.

4. 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.

6. 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.

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

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

11. 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

12. = = 16
Asymptotes:

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

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

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

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

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

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

21. 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)