Table of Contents

Toggle## Introduction

A hyperbola is the locus of a point in a plane which moves in the plane in such a way that the ratio of its distance from a fixed point in the same plane to its distance from a fixed line is always constant, which is always greater than unity.

## Transverse and Conjugate Axes

- The line through the foci of the hyperbola is called the transverse axis.
- The line through the centre and perpendicular to the transverse axis of the hyperbola is called its conjugate axis.

## Hyperbola

## Conjugate Hyperbola

## Asymptote

An asymptote to a curve is a straight line, at a finite distance from the origin, to which the tangent to a curve tends as the point of contact goes to infinity.

(i) The equation of two asymptotes of the hyperbola {x^2\over a^2}-{y^2\over b^2} = 1 are

y = ± b\over a x or x\over a ± y\over b = 0

(ii) The combined equation of the asymptotes of the hyperbola

x^2\over a^2 – y^2\over b^2 = 1 is x^2\over a^2 – y^2\over b^2 = 0(iii) When b = a, i.e asymptotes of rectangular hyperbola x^{2} – y^{2} = a^{2} are y = ± x which are at right angle.

(iv) A hyperbola and its conjugate hyperbola have the same asymptotes.

(v) The asymptotes pass through the centre of the hyperbola.

(vi) The bisector of the angle between the asymptotes of hyperbola {x^2\over a^2}-{y^2\over b^2} = 1 are the coordinate axes.

(vii) The angle between the asymptotes of {x^2\over a^2}-{y^2\over b^2} = 1 is 2tan^{-1} (b/a) or 2sec^{-1} (e).

## Rectangular Hyperbola

A hyperbola whose asymptotes include a right angle is said to be a rectangular hyperbola.

In a hyperbola x^2\over a^2 – y^2\over b^2 = 1, if b = a, then it said to be rectangular hyperbola. The eccentricity of a rectangular hyperbola is always √2.