Hyperbola :
A conic with an eccentricity greater than 1. It has two branches extending up to infinity, tow axes of symmetry, two foci, two directrices, two vertices, and two-axis that are transverses and conjugate and it has a center which is the midpoint of the line of two foci. The transverse axis is the line joining the foci. The Conjugate axis is the line through the center and the transverse axis. Standard equation of the hyperbola is
X^2 /a^2 -y^2 /b^2 =1

Where a and b are the length of semi transverse
conjugate axis respectively, the center being at the
Coordinate axes are along the transverse and con axes. Vertices are (plus/minus a,o), foci (puls/minus ae,o) he eccentricity, where
e= square root of 1+b^2/a^2 the eccentricity. Equations of the directrices are

x=plus/minus (a/e). It has two asymptotes, y = plus, minus [b/a] X. If b is equal to a. It is a rectangular hyperbola with equation x^2-y^2=a^2. If the axis is rotated through an angle the pi over 4 the equation of the rectangular hyperbola will be xy=c^2 in which coordinate axis will be it asymptotes.

Hyperbolic Functions:

Functions having an analogy to circular functions, names hyperbolic sine, hyperbolic cosine etc and defined as

Sinhx= (e^2-e^2)/2
cosh =(e^2+e^2)/2

tanh x=quotient of sinhx and coshx
cothx=quotient of 1 and tanhx
sech x = quotient of 1 and coshx
cosech x = quotient of 1 and sinhx

These are related to circular functions by the relations
sin (ix) =I sinh x,cos (ix) = cosh x etc. Also

cosh^2 x-sinh^2 x is equal to 1
sech^2 x is equal to sech^2 x + tanh^2 x is equal to 1
coth^2 x -cosec^h^2 x is equal to 1
sinh 2x=2 sinh x cosh x
cosh 2x=cosh^2 x + singh^2 x