Hyperbola - Equation, Definition & Properties

Hyperbola is one of the fundamental shapes in geometry formed by intersecting a right circular cone with a plane at an angle such that both halves of the cone are intersected. It is often encountered in both mathematics and real-world applications. Defined as the set of all points where the difference of the distances to two fixed points (called foci) is constant. It is a smooth curve in a plane with two branches that mirror each other, resembling two infinite bows.

Hyperbolas are closely related to ellipses and parabolas, yet they possess distinct properties and applications. From the design of satellite dishes to the paths of celestial bodies, hyperbolas play a critical role in various scientific and engineering fields.

Hyperbola

Hyperbola is the focus of points whose difference in the distances from two foci is constant. This difference is obtained by subtracting the distance of the nearer focus from the distance of the farther focus.

If P (x, y) is a point on the hyperbola and F, F' are two foci, then the locus of the hyperbola is 

PF - PF' = 2a

In analytic geometry, a hyperbola is a type of conic section created when a plane cuts through both halves of a double right circular cone at an angle. This intersection results in two separate, unbounded curves that are mirror images of each other.

Standard Equation of Hyperbola

The standard equations of a hyperbola are:

\bold{\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}= 1}

OR

\bold{\frac{y^{2}}{a^{2}}-\frac{x^{2}}{b^{2}}= 1}

A hyperbola has two standard equations. These equations are based on its transverse axis and conjugate axis. 

• The standard equation of the hyperbola is [(x²/a²) - (y²/b²)] = 1, where the X-axis is the transverse axis and the Y-axis is the conjugate axis. 
• Furthermore, another standard equation is [(y²/a²)- (x²/b²)] = 1, where the Y-axis is the transverse axis and the X-axis is the conjugate axis.
• Standard equation of the hyperbola with center (h, k) and the X-axis as the transverse axis and the Y-axis as the conjugate axis is,

\bold{\frac{(x-h)^{2}}{a^{2}}-\frac{(y-k)^{2}}{b^{2}}= 1}

• Furthermore, another standard equation of the hyperbola with center (h, k) the Y-axis as the transverse axis, and the X-axis as the conjugate axis is

\bold{\frac{(y-k)^{2}}{a^{2}}-\frac{(x-h)^{2}}{b^{2}}= 1 }

Real-Life Application of Hyperbola

• Guitar - The narrow portion of the classical guitar, known as the waist, resembles the shape of a hyperbola, contributing both to its aesthetic appeal and the way it fits comfortably against the player's body.
• Wavefronts - When two stones are thrown into a pool simultaneously, they create concentric ripples that move outward and intersectat point forming Hyperbolas.
• Lenses and Monitors - Objects designed for use with our eyes make heavy use of hyperbolas. These object include microscopes, telescopes and television.
• Satellites - Satellities system make heavy use of hyperbolas and hyorbolic functions. When scientist launch a satellite into space, they must first use mathematical equation to predict its path.

Parts of Hyperbola

Hyperbola is a conic section that is developed when a plane cuts a double right circular cone at an angle such that both halves of the cone are joined. It can be described using concepts like foci, directrix, latus rectum, and eccentricity. The following table represents the parts of the hyperbola:

Eccentricity & Latus Rectum of Hyperbola

Eccentricity

The eccentricity of a hyperbola is the ratio of the distance of a point from the focus to its perpendicular distance from the directrix. It is denoted by the letter 'e'.

• The eccentricity of a hyperbola is always greater than 1, i.e., e >1.
• We can easily find the eccentricity of the hyperbola by the formula :

e = √[1 + (b²/a²)]

where,

• a is the length of the Semi-major axis
• b is the length of the Semi-minor axis

Read More: Eccentricity

Latus Rectum

Latus rectum of a hyperbola is a line passing through any of the foci of a hyperbola and perpendicular to the transverse axis of the hyperbola. The endpoints of a latus rectum lie on the hyperbola, and its length is 2b²/a.

Read more: Latus Rectum

Derivation of Equation of Hyperbola

Let us consider a point P on the hyperbola whose coordinates are (x, y). From the definition of the hyperbola, we know that the difference between the distance of point P from the two foci F and F' is 2a, i.e., PF'-PF = 2a.

Let the coordinates of the foci be F (c, o) and F '(-c, 0).

Now, by using the coordinate distance formula, we can find the distance of point P (x, y) to the foci F (c, 0) and F '(-c, 0).

√[(x + c)² + (y - 0)²] - √[(x - c)² + (y - 0)²] = 2a
⇒ √[(x + c)² + y²] = 2a + √[(x - c)² + y²] 

Now, by squaring both sides, we get

(x + c)² + y² = 4a² + (x - c)² + y² + 4a√[(x - c)² + y²] 
⇒ 4cx - 4a² = 4a√[(x - c)² + y²] 
⇒ cx - a² = a√[(x - c)² + y²] 

Now, by squaring on both sides and simplifying, we get

[(x²/a²) - (y²/(c² - a²))] = 1

We have, c² = a² + b², so by substituting this in the above equation, we get

x²/a² - y²/b² = 1

Hence, the standard equation of the hyperbola is derived.

Hyperbola Formula

Following formulas are widely used in finding the various parameters which include, the equation of hyperbola, the major and minor axis, eccentricity, asymptotes, vertex, foci, and semi-latus rectum.

Where,

• ( x₀ ​, y₀​) is the Centre Point
• a is the Semi-major Axis
• b is the Semi-minor Axis.

Graph of Hyperbola

A hyperbola is a curve that has two unbounded curves that are mirror images of each other. The graph shows that curve in the 2-D plane. We can observe the different parts of a hyperbola in the hyperbola graphs for standard equations given below:

Equation of the Hyperbola	Graph of Hyperbola	Parameters of Hyperbola
\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}= 1		Coordinates of the center: (0, 0) Coordinates of the vertex: (a, 0) and (-a, 0) Coordinates of foci: (c, 0) and (-c, 0) The length of the transverse axis = 2a The length of the conjugate axis = 2b The length of the latus rectum = 2b2/a Equations of asymptotes:  y = (b/a) x and y = -(b/a) x Eccentricity (e) = √[1 + (b2/a2)]
\frac{y^{2}}{a^{2}}-\frac{x^{2}}{b^{2}}= 1		Coordinates of the center: (0, 0) Coordinates of the vertex: (0, a) and (0, -a) Coordinates of foci: (0, c) and (0, -c) The length of the transverse axis = 2b The length of the conjugate axis = 2a The length of the latus rectum = 2b2/a Equations of asymptotes: y = (a/b) x and y = -(a/b) x Eccentricity (e) = √[1 + (b2/a2)]

Conjugate Hyperbola

Conjugate Hyperbola are 2 hyperbolas such that the transverse and conjugate axes of one hyperbola are the conjugate and transverse axis of the other hyperbola respectively.

Conjugate hyperbola of (x² / a²) – (y² /b²) = 1 is,

(x² / a²) - (y² / b²) = 1

Where,

• a is a Semi-major axis
• b is Semi-minor axis
• e is Eccentricity of Parabola
• a² = b² (e² − 1)

Properties of Hyperbola

• If the eccentricities of the hyperbola and its conjugate are e₁, and e₂ then,

(1 / e₁²) + (1 / e₂²) = 1

• Foci of a hyperbola and its conjugate are concyclic and form the vertices of a square.
• Hyperbolas are equal if they have the same latus rectum.

Auxiliary Circles of Hyperbola

Auxiliary Circle is a circle that is drawn with center C and diameter as a transverse axis of the hyperbola. The auxiliary circle of the hyperbola equation is,

x² + y² = a²

Rectangular Hyperbola

A hyperbola with a transverse axis of 2a units and a conjugate axis of 2b units of equal length is called the Rectangular Hyperbola. i.e. in rectangular hyperbola,

2a = 2b

⇒ a = b

The equation of a Rectangular Hyperbola is given as follows:

x² – y² = a²

Note: Eccentricity of Rectangular hyperbola is √2.

Parametric Representation of Hyperbola

The parametric Representation of auxiliary circles of the hyperbola is:

x = a sec θ, y = b tan θ

People Also Read

• Conic Section
• Parabola
• Circle
• Ellipse

Hyperbola Class 11

In Class 11 mathematics, the study of hyperbolas forms a part of the conic sections in analytic geometry. Understanding hyperbolas at this level involves exploring their definition, standard equations, properties, and various elements associated with them.

• NCERT Solutions Class 11 – Chapter 10 Conic Section 
• Class 11 RD Sharma Solutions – Chapter 27 Hyperbola 
• CBSE Class 11 Maths Notes

Conclusion

• Hyperbola is an interesting and essential geometric shape with importance in both theoretical mathematics and practical applications.
• Standard equation of hyperbola and its unique properties, such as eccentricity and asymptotes, to its real-world usage in fields like astronomy and engineering, the hyperbola represents the elegance and usefulness of mathematical concepts.
• Understanding the hyperbola not only enhances our knowledge of conic sections but also equips us with tools to analyze and interpret various phenomena in the natural world. Whether in the design of satellite dishes, the study of orbital paths, or the exploration of mathematical theories, the hyperbola remains a vital and intriguing subject in the study of geometry.
• The properties of hyperbolas make them significant in various fields, including astronomy, physics, and engineering, for modeling and analyzing hyperbolic trajectories and behaviors.

Solved Examples on Hyperbola

Question 1: Determine the eccentricity of the hyperbola x²/64 - y²/36 = 1.

Solution:

Equation of hyperbola is x²/64 - y²/36 = 0

By comparing given equation with standard equation of the hyperbola x²/a² - y²/b² = 1, we get

a² = 64, b² = 36
⇒ a = 8, b = 6

We have,

Eccentricity of a hyperbola (e) = √(1 + b²/a²)

⇒ e = √(1 + 6²/8²)
⇒ e = √(1 + 36/64)
⇒ e = √(64 + 36)/64) = √(100/64)
⇒ e = 10/8 = 1.25‬

Hence, Eccentricity of given hyperbola is 1.25‬.

Question 2: If the equation of the hyperbola is [(x-4)²/25] - [(y-3)²/9] = 1, find the lengths of the major axis, minor axis, and latus rectum. 

Solution:

Equation of hyperbola is [(x-4)²/25] - [(y-3)²/9] = 1

By comparing given equation with the standard equation of the hyperbola, (x - h)²/a² - (y - k)²/b² = 1

Here, x = 4 is the major axis and y = 3 is the minor axis.

a² = 25 a = 5
b² = 9 b = 3

Length of major axis = 2a = 2 × (5) = 10 units
Length of minor axis = 2b = 2 × (3) = 6 units
Length of latus rectum = 2b²/a = 2(3)²/5 = 18/5 = 3.6 units

Question 3: Find the vertex, asymptote, major axis, minor axis, and directrix if the hyperbola equation is [(x-6)²/7²]-[(y-2)²/4²] = 1.

Solution:

Equation of hyperbola is [(x-6)²/7²] - [(y-2)²/4²] = 1

By comparing given equation with standard equation of hyperbola, (x - h)²/a² - (y - k)²/b² = 1

h = 6, k = 2, a = 7, b = 4

Vertex of a Hyperbola: (h + a, k) and (h - a, k) = (13, 2) and (-1, 2)
Major axis of Hyperbola is x = h x = 6
Minor axis of Hyperbola is y = k y = 2

Equations of asymptotes of hyperbola are 

y = k − (b / a)x + (b / a)h and y = k+ (b / a)x - (b / a)h
⇒ y = 2 - (4/7)x + (4/7)6 and y = 2 + (4/7)x - (4/7)6
⇒ y = 2 - 0.57x + 3.43 and y = 2 + 0.57x - 3.43
⇒ y = 5.43 - 0.57x and y = -1.43 + 0.57x

Equation of the directrix of a hyperbola is x = ± a²/√(a² + b²)

⇒ x = ± 7²/√(7² + 4²) 
⇒ x= ± 49/√65 
⇒ x = ± 6.077

Question 4: Find the eccentricity of the hyperbola whose latus rectum is half of its conjugate axis.

Solution:

Length of latus rectum is half of its conjugate axis

Let,
Equation of hyperbola be [(x² / a²) – (y² / b²)] = 1

Conjugate axis = 2b
Length of Latus rectum = (2b² / a)

From given data,
(2b² / a) = (1/2) × 2b
2b = a

We have,

Eccentricity of Hyperbola (e) = √[1 + (b²/a²)]

Now, substitute a = 2b in the formula of eccentricity

⇒ e = √[1 + (b²/(2b)²]
⇒ e = √[1 + (b²/4b²)] = √(5/4) 
⇒ e = √5/2

Hence, required eccentricity is √5/2.

Practice Problems on Hyperbola

Question 1: Find the standard form equation of the hyperbola with vertices at (-3, 2) and (1, 2), and a focal length of 5.

Question 2: Determine the center, vertices, and foci of the hyperbola with the equation 9x² - 4y² = 36.

Question 3: Given the hyperbola with the equation (x - 2)²/16 - (y + 1)²/9 = 1, find the coordinates of its center, vertices, and foci.

Question 4: Write the equation of the hyperbola with a horizontal major axis, center at (0, 0), a vertex at (5, 0), and a focus at (3, 0).