HYPERBOLA FORMULA

HYPERBOLA FORMULA

In simple sense, hyperbola looks similar to to mirrored parabolas. The two halves are called the branches. When the plane intersect on the halves  of a right circular cone angle of which will be parallel to the axis of the cone, a parabola is formed.

A hyperbola contains: two foci and two vertices. The foci of the hyperbola are away from the hyperbola’s center and vertices. Here is an illustration to make you understand:

Hyperbola Formula

The equation for hyperbola is,

(x−x0)2a2−(y−y0)2b2=1

Where,

x0,y0
are the center points.
a
= semi-major axis.
b
= semi-minor axis.

Let us learn the basic terminologies related to hyperbola formula:

MAJOR AXIS

The line that passes through the center, focus of the hyperbola and vertices is the Major Axis. Length of the major axis = 2a. The equation is given as:

y=y0

MINOR AXIS

The line perpendicular to the major axis and passes by the middle of the hyperbola is the Minor Axis.
Length of the minor axis = 2b. The equation is given as:

x=x0

ECCENTRICITY

The variation in the conic section being completely circular is eccentricity. It is usually greater than 1 for hyperbola. Eccentricity is

22
for a regular hyperbola. The formula for eccentricity is:

a2+b2a

ASYMPTOTES

Two bisecting lines that is passing by the center of the hyperbola that doesn’t touch the curve is known as the Asymptotes. The equation is given as:

y=y0+bax−bax0

y=y0−bax+bax0

Directrix of a hyperbola

Directrix of a hyperbola is a straight line that is used in generating a curve. It can also be defined as the line from which the hyperbola curves away from. This line is perpendicular to the axis of symmetry. The equation of directrix is:

x=±a2a2+b2

VERTEX

The point of the branch that is stretched and is closest to the center is the vertex. The vertex points are

(a,y0) and (−a,y0)

Focus (foci)

On a hyperbola, focus (foci being plural) are the fixed points such that the difference between the distances are always found to be constant. The two focal points are: 

(x0+a2+b2,y0)

(x0−a2+b2,y0)

Solved examples

Question: The equation of the hyperbola is given as: 

(x−4)292−(y−2)272
 

Find the following: Vertex, Asymptote, Major Axis, Minor Axis and Directrix?

Solution:

Given,

x0=4
y0=2
a=9
b=7

The vertex point: 

(a,y0)
and 
(−a,y0)
are
(9,2)
and
(−9,2)

Asymptote

y=79(x−4)+2y=−79(x−4)+2

Major Axis

a = 9

Minor Axis

b = 7

Directrix

x=±9292+72=±8181+49=7.1

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