Information on Hyperbolas

Hyperbolas
A hyperbola is the set of all points in a plane whose distances from two fixed points in the plane have a constant difference. The fixed points are the foci of the hyperbola. The line through the foci is the focal axis. The point on the focal axis midway between the foci is the center. The points where the hyperbola intersects its focal axis are the vertices of the hyperbola.
The standard for of the hyperbola equation is: x2/a2 – y2/b2 = 1 and this is for when the hyperbola is centered at the origin with the x-axis as its focal axis. A hyperbola centered at the origin with the y-axis as its focal axis is the inverse relation of the earlier formula, and is thus in the form: y2/a2 - x2/b2 = 1.
A line segment with endpoints on a hyperbola is a chord of the hyperbola. The chord lying on the focal axis connecting the vertices is the transverse axis of the hyperbola. The length of the transverse axis is 2a. The line segment of length 2b that is perpendicular to the focal axis and that has the center of the hyperbola as its midpoint is the conjugate axis of the hyperbola. The number a is the semitransverse axis, and b is the semiconjugate axis.

Hyperbolas with Center (0, 0)
Standard Equation

Focal Axis x-axis y-axis
Foci
( ±c, 0)
(0, ±c)
Vertices
(±a, 0)
(0, ±a)
Semitransverse axis a a
Semiconjugate axis b b
Pythagorean relation

Asymptotes

Hyperbolas with Center (h, k)

Standard Equation

Focal Axis y=k x=h
Foci
( h ± c, k)
(h, k ± c)
Vertices
(h ± a, k)
(h, k ± a)
Semitransverse axis a a
Semiconjugate axis b b
Pythagorean relation

Asymptotes

See pg. 656-658 in textbook for drawings.