Definition (1): A hyperbola is the set of points in a plane, the distance of each of which from the focus is e ( > 1) times its distance from the directrix.
Definition (2): If the eccentricity (e) is greater than 1 then the conic is called a hyperbola.
CONCEPTUAL THEOREM
1. Show that the equation of the hyperbola in the standard form is
Proof:
Let 'S' be the focus
Let 'l' be the directrix and SZ the perpendicular from S on the directrix.
Let A and A' are points on SZ such that
Let P(x, y) be any point on the hyperbola.
By definition of hyperbola: 
PS = ePM
x2 + a2e2 - 2axe + y2 = x2e2 + a2 − 2axe
x2 + a2e2 + y2 = x2e2 + a2
x2 (1 - e2) + y2 = a2 (1-e2)
for e > 1, there exists a real number 'b' ; such that b2 = a2 (e2 - 1)
Which is the required standard form of the hyperbola.
DIFFERENT FORMS OF HYPERBOLA
(a > b) then
Centre : C (0, 0)
Foci : (± ae, 0)
Vertices : (± a, 0)
Directrices : x = ± ( 
)
Axis : x − axis (y = 0)
Transverse Axis : y = 0
Conjugate Axis : y − axis (x = 0)
Length of Transverse Axis : 2a
Length of conjugate axis : 2b
Equation of latus rectum : x = ± ae
Centre : C' (0, 0)
Foci : (∆, ± be)
Vertices : (0, ± b)
Directrices : y = ± b/e
Axis : y − axis (x = 0)
Transverse axis : x = 0
Conjugate axis : x − axis (y = 0)
Length of transverse axis : 2b
Length of conjugate axis : 2a
Equation of latus rectum : y = ± be
[... a2 = b2 (e2 - 1)]
Centre : C (α, β)
Foci : (α ± ae, β)
Vertices : (α ± a, β)
Directrices : x = α ± 
Axis : y = β
(T.A.) Transverse Axis : y- β = 0
(C.A.) Conjugate Axis : x- α = 0
Length of transverse axis (L.T.A.) : 2a
Length of conjugate axis (L.C.A.) : 2b
Equation of latus rectum : x = α ± ae
Centre : C (α, β)
Foci : (α, β ± be)
Vertices : (α, β ± b)
Directrices : y = β ± b/e
Axis : x = α
Transverse axis (T.A.) : x - α = 0
Conjugate axis (C.A.) : y - β = 0
Length of transverse axis (L.T.A) : 2b
Length of conjugate axis (L.C.A) : 2a
Equation of latus rectum : y = β ± be
