# Hyperbola Formula | Class 11 Mathematics

## 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

 Equation of hyperbola {x^2\over a^2}-{y^2\over b^2} = 1 Vertex A(a, 0) and A'(-a, 0) Centre O (0, 0) Length of the transverse axis 2a Length of the conjugate axis 2b Foci F(ae, 0) and F'(- ae, 0) The equation of directories are x = a\over e and x = – a\over e Length of the latus rectum 2b^2\over a Eccentricity (e) \sqrt {(1+{b^2\over a^2})} or b2 = a2(e2 – 1) Distance between foci 2ae Distance between directrices 2a\over e Coordinate of end of latus rectum (± ae, ± b^2\over a)

## Conjugate Hyperbola

 Equation of hyperbola -{x^2\over a^2}+{y^2\over b^2} = 1 Vertices A (0, b) and A’ (0, -b) Centre O (0, 0) Length of the transverse axis 2a Length of the conjugate axis 2b Foci F (0, be) and F'(0, – be) The equation of directories are x = b\over e and x = – b\over e Length of the latus rectum 2a^2\over b Eccentricity (e) \sqrt {(1+{a^2\over b^2})} or b2 = a2(e2 – 1) Distance between foci 2be Distance between directrices 2b\over e Coordinate of end of latus rectum (± a^2\over b, ± be)

## 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^2y^2\over b^2 = 1 is x^2\over a^2y^2\over b^2 = 0

(iii) When b = a, i.e asymptotes of rectangular hyperbola x2 – y2 = a2 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^2y^2\over b^2 = 1, if b = a, then it said to be rectangular hyperbola. The eccentricity of a rectangular hyperbola is always √2.

 Equation x2 – y2 = a2 Asymptotes x ± y = 0 Eccentricity (e) √2 Centre (0, 0) Foci F(√2a, 0) and F'(- √2a, 0) Vertices A (a, 0) and A'(-a, 0) Equation of directrices x = a\over √2 and x = – a\over √2 Length of the latus rectum 2a Paramatic form x = a sec θ, y = a tan θ Equation of tangent x sec θ – y tan θ = a Equation of normal x\over sec θ + y\over tan θ

 Equation xy = c2 Asymptotes x = 0 and y = 0 Eccentricity (e) √2 Centre O (0, 0) Foci F(√2c, √2c) and F'(- √2c, -√2c) Vertices A (c, c) and A'(-c, c) Equation of directrices x + y = ± √2 c Length of the latus rectum 2√2 c Paramatic form x = c t, y = c\over t
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