Hyperbola

Hyperbola is one of the most important conic sections. A hyperbola is obtained by cutting a right circular cone at both nappe by a plane. In simple words, a hyperbola can be defined as the locus of a point in a plane which moves in such a way that its distance from a fixed point is ‘e’ times the distance from the fixed straight line where ‘e’ is the eccentricity. In case of hyperbola, the eccentricity is always greater than 1. 


As shown in the figure above, the hyperbola is symmetrical about two axes and one of the branches can be considered to be the reflection of the other about the axes. The concepts of hyperbola are quite identical to the concepts of ellipse; hence students are advised to go through the section of ellipse first before the one on hyperbola. Most of the results of hyperbola can be derived from those of ellipse by changing the sign of a single term.

 

Some of the topics covered in the chapter include:

 

We will just give an outline of these topics in this section without going into the intricacies as they have been discussed in detail in the coming sections.

Directrix:  

The directrix of a hyperbola may be defined as the line segment from which the distance of the hyperbola is always proportional to the distance from the focus. 


 

 

Focus of a hyperbola:

A hyperbola has two foci which are S(ae, 0) and S’ = (-ae, 0).

 

Transverse and Conjugate axes:

The points A (a, 0) & A’ (–a, 0) are called the vertices of the hyperbola and the line AA’ joining the vertices is called the transverse axis and the line perpendicular to it, through the centre (0, 0) of the hyperbola is called conjugate axis.

 

Relation between focal distances:

The difference of the focal distances of any point on the hyperbola is constant and equal to transverse axis.

 

Conjugate Hyperbolas:

Two hyperbolas are said to be conjugate of one another if the transverse and conjugate axis of one are respectively the conjugate and transverse axis of the other.

Eg: x2/a2 – y2/b2 = 1 and –x2/ a2 + y2/b2 = 1 are conjugate hyperbola of each. Various properties regarding the eccentricity of conjugate hyperbola have been discussed in the following sections.

 

Similar Hyperbolas:

Hyperbolas are always judged according to their eccentricity and hence if two hyperbolas have the same eccentricity they are said to be similar.

 

For more on hyperbola,you may view the following video

 

Parametric coordinates:

In parametric coordinates, the hyperbola x2/a2 – y2/b2 = 1 can be represented by the equations x = a sec θ and y = b tan θ, where θ is the parameter. The parametric equations x = a cosh φ and y = b sinh φ, also represent the same hyperbola.

 

Asymptotes:

If the length of the perpendicular drawn from a point on the hyperbola to a straight line tends to zero as the point on the hyperbola is made to approach infinity, then the straight line is called the asymptote of hyperbola.

For more on asymptotes of rectangular hyperbola or asymptotes of conjugate hyperbola, please refer the following sections.

 

Rectangular Hyperbola:

The general equation of the rectangular hyperbola in parametric coordinates is x = ct, y = c/t, t ∈ R-{0}.

Hyperbola is an important topic of coordinate geometry. It is important for the students to have clarity in the concepts of hyperbola in order to remain competitive in the JEE. Since the concepts are quite related to those of ellipse, hence gaining expertise in ellipse can help you master hyperbola as well.      

 

 Related resources:

Related Resources
Examples on Propositions of Hyperbola

Examples on Propositions of Hyperbola...

Relation between Focal Distances

Relation between Focal Distances The difference of...

Examples Based on Hyperbola

Examples Based on Hyperbola Illustration: Find the...

Ellipse Vs Hyperbola

Ellipse Vs Hyperbola Conic Sections is an...

Solved Examples on Hyperbola

Solved Examples on Hyperbola Example: Show that...

Rectangular Hyperbola

Rectangular Hyperbola What do we mean by a...

Basic Concepts of Hyperbola

Basic Concepts of Hyperbola A hyperbola is the...

Intersection of Circle and Rectangular Hyperbola

Intersection of a Circle and a Rectangular...

Parametric Coordinates

Parametric Coordinates We can express the...

Examples on Finding Locus of Point

Examples on Finding Locus of Point Example: C is a...

Important Properties of Hyperbola

Important Properties of Hyperbola Hyperbola is an...

Conjugate Hyperbola

Conjugate hyperbola What do you mean by a...

Propositions of Hyperbola

Propositions of a Hyperbola What is a hyperbola? A...