Hyperbola

Definition :-A hyperbola is the set of all points in a plane, the difference of whose

distances from two fixed points in the plane is a constant.



The term “difference” that is used in the definition means the distance to the

farther point minus the distance to the closer point. The two fixed points are called the

foci of the hyperbola. The mid-point of the line segment joining the foci is called the

centre of the hyperbola. The line through the foci is called the transverse axis and

the line through the centre and perpendicular to the transverse axis is called the conjugate

axis. The points at which the hyperbola

intersects the transverse axis are called the

vertices of the hyperbola (Fig 11.29).

We denote the distance between the

two foci by 2c, the distance between two

vertices (the length of the transverse axis)

by 2a and we define the quantity  b as

b = 2 2

c –a

Also 2b is the length of the conjugate axis

(Fig 11.30).




To find the constant P1

F2

 – P1

F1 :

By taking the point P at A and B in the Fig 11.30, we have

BF1 – BF2 = AF2

 – AF1

 (by the definition of the hyperbola)

BA +AF1

– BF2 = AB + BF2

– AF1

i.e., AF1 = BF2

So that,

BF1

 – BF2 = BA + AF1

– BF2

 = BA = 2a