Hyperbola Formula: Definition, Equation & Solved Examples

Content Curator

Hyperbola is a type of curve that lies in a plane and has two pieces that are mirror images of each other. These two mirror images are called connected components and resemble two bows. It is an important conic section in mathematics, formed by the intersection of the double cone by a plane surface, not necessarily at the centre.

Table of Content

What is Hyperbola?
Hyperbola Equation
Solved Examples of Hyperbola
Things to Remember
Previous Year Questions
Sample Questions

Key Terms: Hyperbola, Eccentricity, Curve, Cone, Foci, Locus, Plane, Vertex, Angle

What is Hyperbola?

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In geometry, a conic section formed when a plane intersects a double right circular cone at such an angle that both the halves of the cone are intersected, is known as a hyperbola. It is a set of points where the difference in distances from two foci is a constant value and is calculated as the difference between the distance of the farther point and the nearer point.

For example, for a point P(x, y) on the hyperbola and for two foci F, F', the locus of the hyperbola is PF - PF' = 2a.

Hyperbola

Hyperbola

Hyperbola Equation

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The equation of the hyperbola is given as:

(x−x₀)²/a² – (y-y₀)²/b² = 1

Where,

x₀, y₀ = the centre points
a = semi-major axis
b = semi-minor axis

The general equation for a hyperbola is given as:

x²/a² – y²/b² = 1

Here,

x is the transverse axis of hyperbola
y is the conjugate axis of hyperbola

The other formulas/equations of hyperbola and its terminologies are briefed below.

Minor Axis

Minor axis is the line perpendicular to the major axis and crosses through the hyperbola's centre. The length of the minor axis is 2b and the equation is as follows:

x = x₀

Major Axis

Major axis is the line that crosses the middle, the focus of the hyperbola and the vertices. The length of the major axis is 2a and its equation is given as:

y = y₀

Eccentricity

Eccentricity is the differentiation in the conic section being fully circular. For a point on a hyperbola, eccentricity is the ratio of its distance from the focus and the directrix.

Generally, it is higher than 1 for hyperbola and is 2√2 for a regular hyperbola. The formula for eccentricity is:

(√a² + b²)/a

Asymptotes

Asymptotes are two intersecting line segments which cross through the centre of the hyperbola and do not touch the curve. The formula for asymptotes is given as:

y = y₀ + b/ax – b/ax₀

y = y₀ − b/ax + b/ax₀

Directrix of Hyperbola

The directrix of a hyperbola is a straight line which is used to generate a curve. It is the line segment the hyperbola curves away from and is perpendicular to the axis of symmetry. The equation of the directrix formula is given as:

X = a²/(√a² + b²)

Vertex

Vertex is the point of the branch that is stretched and is closest to the centre. The vertex points are given as:

(a, y₀) and (-a, y₀)

Focus(Foci)

Focus (foci as plural) are the fixed points such that the difference between the distances is always found to be constant. The two focal points are given as:

[x₀ + (√a² + b²), y₀] and [x₀ - (√a² + b²), y₀]

Also Read:

Related Concepts
Parabola Formula	Hyperbolic Function Formula	Ellipse Formula
Parabola Graph	Circles	Area of a Circle

Solved Examples of Hyperbola

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To understand better the concept of hyperbola, a few examples are illustrated below.

Example 1: The equation of hyperbola is given as (x - 4)²/9² - (y - 2)²/7²

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

Solution: Given

x₀ = 4, y₀ = 2 and a = 9, b = 7

Therefore, the vertex is (9, 2) and (-9, 2)

Asymptote is: y = 7/9(x - 4) + 2 and y = -7/9(x - 4) + 2
Major axis is 9 and minor axis is 7.
Directrix = 9²/(√9² + 7²) = 81/(√81 + 49) = 7.1

Example 2: The equation of the hyperbola is given as (x - 5)²/4² - (y - 2)²/ 2² = 1. Find the length of the Major Axis and Minor Axis.

Solution: According to the formulas:

Length of the major axis = 2a, and the length of the minor axis = 2b
Length of the major axis = 2 × 4 = 8, and the length of the minor axis = 2 × 2 = 4

Things to Remember

A hyperbola can be defined as a type of curve which lies in a plane and has two pieces that are mirror images of each other.
Hyperbola is a set of points where the difference in distances from two foci is a constant value.
Minor axis of a hyperbola is the line perpendicular to its major axis and it crosses through the hyperbola's centre.
Eccentricity, for a point, is defined as the ratio of its distance from the focus and the directrix.

Previous Year Questions

Let P (4, 3) be a point on the hyperbola… [WBJEE 2019]
If e and e′ are the eccentricities of hyperbola… [BCECE 2007]
For the hyperbola… [BITSAT 2007]
The length of the straight line… [VITEEE 2007]
The equation of the hyperbola with vertices (3,0), (-3,0)... [VITEEE 2018]
The foci of a hyperbola coincide with the foci of the ellipse… [JEE Mains 2018]
The slope of the tangent to the hyperbola… [BITSAT 2010]
The distance between the foci of a hyperbola is 16 and its eccentricity… [KCET 2018]
The locus of the point of intersection of the lines… [JEE Mains 2018]
The eccentricity of the hyperbola with latus rectum… [AMUEEE 2011]

Sample Questions

Ques: The equation of the hyperbola is given as [(x - 5)²/6²] - [(y - 2)²/ 4²] = 1. Use the hyperbola formulas to find the length of the Major Axis and Minor Axis. (2 Marks)

Ans: The hyperbola formula for the length of the major and minor axis is given as;

Length of major axis = 2a, and length of minor axis = 2b

Length of major axis = 2 × 6 = 12, and Length of minor axis = 2 × 4 = 8.

Ques: What is the Latus Rectum of the hyperbola? (1 Mark)

Ans: Latus rectum of a hyperbola is defined as a line perpendicular to the transverse axis via any of the foci with its endpoints lying on the hyperbola. In a hyperbola, the length of the latus rectum is 2b/a.

Ques: The equation of the hyperbola is given as (x - 3)²/5² - (y - 2)²/ 4² = 1. Find the asymptote of this hyperbola. (3 Marks)

Ans: The formula for the asymptotes of a hyperbola is:

y = y₀ + b/ax – b/ax₀

y = y₀ − b/ax + b/ax₀

Using the above formula

y = 2 - (4/5)x + (4/5)5 and y = 2 + (4/5)x - (4/5)5

Asymptotes are y = 2 - (4/5)x + 4, and y = 2 + (4/5)x - 4.

Ques: What is hyperbola in a conic section? (1 Mark)

Ans: In a conic section, a hyperbola is a curve that lies in a plane and has two pieces that are mirror images of each other and are called connected components and resemble two bows.

Ques: What is the general formula of a hyperbola? (2 Marks)

Ans: The general equation for a hyperbola is given as x 2/a² – y²/b² = 1

Where x is the transverse axis of the hyperbola and y is the conjugate axis of the hyperbola.

Ques: Define a rectangular hyperbola. (1 Mark)

Ans: The hyperbola whose major axis and minor axis are of equal length, is called a rectangular hyperbola. Hence we have 2a = 2b, or a = b. The equation of the rectangular hyperbola is x² - y² = a².

Ques: What is the eccentricity of a hyperbola? (2 Marks)

Ans: For a point on a hyperbola, eccentricity can be defined as the ratio of its distance from the focus and the directrix.

Generally, it is higher than 1 for hyperbola and is 2√2 for a regular hyperbola. The formula for eccentricity is: (√a² + b²)/a

Ques: The equation of the hyperbola is given as (x - 2)²/9² - (y - 3)²/4², find its vertices. (1 Mark)

Ans: Here, x₀ = 2, y₀ = 3 and a = 9, b = 4 and so the vertex will be (9, 3) and (-9, 3).

Ques: What is the conjugate axis of a hyperbola? (1 Mark)

Ans: The axis line which passes through the centre of the hyperbola and is perpendicular to its transverse axis is called the conjugate axis of the hyperbola.

Ques: Define the transverse axis. (1 Mark)

Ans: The transverse axis of a hyperbola is the line which passes through the centre and the two foci of the hyperbola.

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Hyperbola Formula: Definition, Equation & Solved Examples

What is Hyperbola?

Hyperbola Equation

Minor Axis

Major Axis

Eccentricity

Asymptotes

Directrix of Hyperbola

Vertex

Focus(Foci)

Solved Examples of Hyperbola

Things to Remember

Previous Year Questions

Sample Questions

CBSE CLASS XII Related Questions

1.
Evaluate:
\[ \int_{0}^{1} \frac{x \tan^{-1}x}{(1+x^2)^{3/2}}\,dx \]

3.
Find:

The principal value of \[ \sec^{-1}(\sqrt{2})+2\csc^{-1}(-2) \] is:

5.
A relation $R$ on set $A=\{1,2,3\}$ defined as $R=\{(1,2),(2,1),(2,2)\}$ is

6.
Find:

If the function \[ f(x)= \begin{cases} \frac{\sin x}{x}+\cos x, & x\neq0\\ k, & x=0 \end{cases} \] is continuous at \(x=0\), then find the value of \(k\).

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Hyperbola Formula: Definition, Equation & Solved Examples

What is Hyperbola?

Hyperbola Equation

Minor Axis

Major Axis

Eccentricity

Asymptotes

Directrix of Hyperbola

Vertex

Focus(Foci)

Solved Examples of Hyperbola

Things to Remember

Previous Year Questions

Sample Questions

CBSE CLASS XII Related Questions

1.Evaluate:\[ \int_{0}^{1} \frac{x \tan^{-1}x}{(1+x^2)^{3/2}}\,dx \]

3. Find: The principal value of \[ \sec^{-1}(\sqrt{2})+2\csc^{-1}(-2) \] is:

5.A relation $R$ on set $A=\{1,2,3\}$ defined as $R=\{(1,2),(2,1),(2,2)\}$ is

6. Find: If the function \[ f(x)= \begin{cases} \frac{\sin x}{x}+\cos x, & x\neq0\\ k, & x=0 \end{cases} \] is continuous at \(x=0\), then find the value of \(k\).

Comments

SUBSCRIBE TO OUR NEWS LETTER

1.
Evaluate:
\[ \int_{0}^{1} \frac{x \tan^{-1}x}{(1+x^2)^{3/2}}\,dx \]

3.
Find:

The principal value of \[ \sec^{-1}(\sqrt{2})+2\csc^{-1}(-2) \] is:

5.
A relation $R$ on set $A=\{1,2,3\}$ defined as $R=\{(1,2),(2,1),(2,2)\}$ is

6.
Find:

If the function \[ f(x)= \begin{cases} \frac{\sin x}{x}+\cos x, & x\neq0\\ k, & x=0 \end{cases} \] is continuous at \(x=0\), then find the value of \(k\).