Hyperbola

1.0What is Hyperbola?

The hyperbola is a conic section formed by the intersection of a plane with both halves of a double cone. It consists of two distinct branches, each extending infinitely, and is defined by its geometric properties, including its foci, asymptotes, and eccentricity. Hyperbolas have significant applications in fields such as physics, engineering, astronomy, and economics, making them essential objects of study in mathematics.

In mathematics, a hyperbola is a significant conic section resulting from the intersection of a double cone by a plane surface, albeit not necessarily at its center. Characterized by its symmetry along the conjugate axis, hyperbola bears resemblance to an ellipse. Key concepts like foci, directrix, latus rectum, and eccentricity are applicable to hyperbolas. Here, we aim to comprehend the definition, formula derivation, and standard forms of hyperbolas through illustrated examples.

2.0Hyperbola Definition

A hyperbola, a smooth curve within a plane, comprises two mirrored branches akin to infinite bows. It's defined as a set of points where the difference in distances from two foci remains constant. This difference, taken from the farther focus and then from the nearer focus, yields the locus equation for a point P (x, y) on the hyperbola, with foci F and F'. The equation is given by |PF – PF' |= 2a.

Hyperbola is the locus of a point which moves such that ratio of its distance from a fixed point and a fixed line is constant(>1)

3.0Parts of a Hyperbola

A hyperbola consists of several key components:

1. Vertices: These are the endpoints of the transverse axis and are located on the hyperbola's branches.
2. Foci (Focus singular): The fixed points inside the hyperbola, used in its definition. The distance from any point on the hyperbola to each focus differs by a constant value.
3. Center: The midpoint of the segment connecting the vertices. It's the point where the axes intersect.
4. Transverse Axis: The line segment passing through the vertices, perpendicular to the conjugate axis. It defines the width of the hyperbola.
5. Conjugate Axis: The line segment perpendicular to the transverse axis, passing through the center.
6. Asymptotes: Straight lines that the hyperbola approaches but never touches. They intersect at the center of the hyperbola.
7. Directrices (Directrix singular): These are lines equidistant from the foci, located outside the hyperbola.

Understanding these parts helps in defining and analyzing the properties of a hyperbola in mathematical contexts and real-world applications.

4.0Hyperbola Equation

The standard form of the equation for a hyperbola centered at the origin is:

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

Or

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

depending on whether the hyperbola is oriented horizontally or vertically. Here, a and b are the lengths of the semi-major and semi-minor axes, respectively.

If the hyperbola is translated from the origin to a point (h, k), the equation becomes:

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

Or

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

for horizontal and vertical hyperbolas, respectively.

5.0Eccentricity of Hyperbola

The eccentricity e of a hyperbola can be defined as the ratio of the distance between the center and one of the foci to the distance between the center and either vertex. 

For a hyperbola with the equation:

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

the eccentricity is given by:

$e=\sqrt{1+\frac{b^2}{a^2}}$

The eccentricity of a hyperbola determines its shape. It is always greater than 1.

Hyperbola eccentricity formula provides a quantitative measure of how "stretched out" the hyperbola is. As ‘e’ increases, the hyperbola becomes more elongated.

6.0Asymptote Of Hyperbola

The asymptotes of a hyperbola are straight lines that the hyperbola approaches but never intersects. 

For a hyperbola with the equation:

$\frac{x^2}{a^2}-\frac{y^2}{b^2}=1\text{ or }\frac{y^2}{b^2}-\frac{x^2}{a^2}=1$

the equations of the asymptotes are:

$y=\pm \frac{b}{a}x$

These asymptotes pass through the center of the hyperbola. They provide a visual guide to the behavior of the hyperbola as it extends towards infinity.

7.0Rectangular Hyperbola

A rectangular hyperbola is a type of hyperbola where the asymptotes are perpendicular to each other, resulting in a shape that resembles a rectangle. In other words, the slopes of the asymptotes are opposite reciprocals of each other. 

The equation of a rectangular hyperbola in standard form is:

xy = c²

or

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

where a = b, and c is a constant. 

Rectangular hyperbolas have unique properties and applications in various fields, including optics, physics, and engineering. They are often used in the design of certain types of antennas, lenses, and reflectors due to their symmetrical and predictable characteristics.

8.0Length of Latus Rectum of Hyperbola

The length of the latus rectum (LR) of a hyperbola depends on its eccentricity e and is calculated using the formula:

$\mathrm{LR}=\frac{2{\mathrm{b}}^2}{\mathrm{a}}$

where:

• a is the length of the semi-major axis,
• b is the length of the semi-minor axis.

Alternatively, if the hyperbola is defined by its eccentricity e, the formula for the length of the latus rectum can be expressed as:

L.R = 2a(e² –1)

These formulas provide the length of the line segment perpendicular to the transverse axis and passing through one of the foci. The latus rectum is a significant parameter used to characterize the shape and size of a hyperbola.

9.0Conjugate Axis of Hyperbola

The conjugate axis of a hyperbola is the line segment passing through the center of the hyperbola and perpendicular to the transverse axis. It is the axis of symmetry for the hyperbola 

In the standard form of the equation for a hyperbola:

$\frac{x^2}{a^2}-\frac{y^2}{b^2}=1\text{ or }\frac{y^2}{b^2}-\frac{x^2}{a^2}=1$

where ‘a’ and ‘b’ are the lengths of the semi-transverse and semi-conjugate axes, respectively.

The conjugate axis is perpendicular to the transverse axis and passes through the center of the hyperbola. It is used in calculating various properties of the hyperbola, such as the foci, vertices, and asymptotes.

10.0Transverse Axis of Hyperbola

The transverse axis of a hyperbola is the line segment passing through the center of the hyperbola and containing the vertices.

In the standard form of the equation for a hyperbola:

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

Transverse axis has length 2a.

The transverse axis determines the width of the hyperbola and is used in calculating various properties of the hyperbola, such as the foci, vertices, and asymptotes. It is perpendicular to the conjugate axis, which is the axis of symmetry for the hyperbola.

11.0Hyperbola Solved Examples

Example 1: Find the equation of the hyperbola with foci at (3, 0) and (–3, 0) and eccentricity $=\sqrt{2}$ the x-axis.

Solution: Given that the foci are (3, 0) and (–3, 0) and the vertices are on the x-axis, we can determine that the center of the hyperbola is at the origin (0, 0) because it lies halfway between the foci.

Since the foci are on the x-axis, the transverse axis is horizontal. Therefore, the equation of the hyperbola is of the form:

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

ae = 3

$e=\sqrt{2}$

$\Rightarrow \mathrm{a}=\frac{3}{\sqrt{2}}=\mathrm{b}$

So, the equation of the hyperbola is:

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

$\frac{x^2}{\frac{9}{2}}-\frac{y^2}{\frac{9}{2}}=1$

2(x² – y²) = 9

Example 2: Find the equation of the hyperbola with foci at (5, 0) and (–5, 0), and vertices at (3, 0) and (–3, 0).

Solution: Given that the foci are (5, 0) and (–5, 0), and the vertices are at (3, 0) and (–3, 0), we can determine that the center of the hyperbola is at the origin (0, 0) because it lies halfway between the foci. Since the foci are on the x-axis, the transverse axis is horizontal. Therefore, the equation of the hyperbola is of the form:

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

The distance from the center to each focus is c = 5, and the distance from the center to each vertex is a = 3 (since the vertices are on the x-axis).

We know that for a hyperbola, c² = a² + b². Substituting the given values:

⇒ 5² = 3² + b²

⇒ 25 = 9 + b²

⇒ b² = 16

Therefore, b = 4. 

So, the equation of the hyperbola is:

$\frac{x^2}{9}-\frac{y^2}{16}=1$

Example 3: Find the equation of the hyperbola with foci at (0, 2) and (0, –2), and vertices at (0, 3) and (0, –3).

Solution : Given that the foci are (0, 2) and (0, –2), and vertices at (0, 3) and (0, –3), we can determine that the center of the hyperbola is at (0, 0) because it lies halfway between the foci. Since the foci are on the y-axis, the transverse axis is vertical. Therefore, the equation of the hyperbola is of the form:

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

The distance from the center to each focus is c = 2, and the distance from the center to each vertex is a = 3 (since the vertices are on the y-axis).

We know that for a hyperbola, c² = a² + b². Substituting the given values:

⇒ 2² = 3² + b²

⇒ 4 = 9 + b²

⇒ b² = –5 

But b² cannot be negative, so there is no real solution for b.

Therefore, no such Hyperbola exists.

Example 4: The distance between the foci of hyperbola $\frac{y^2}{25^2}-\frac{x^2}{16^2}=1$ is ? 

Solution :  a² = 25, b² = 16

a² e² = a² + b²

25e² = 41

$e=\frac{\sqrt{41}}{5}$

Distance between foci = 2ae

=$2\times 5\times \frac{\sqrt{41}}{5}$

$I=2\sqrt{41}$

12.0Solved Questions from Hyperbola

Q. What are the standard equations of a hyperbola?

Ans: The standard equations of a hyperbola are:

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

where ‘a’ and ‘b’ are positive constants representing the lengths of the semi-transverse and semi-conjugate axes, respectively.

Q. What is the latus rectum of a hyperbola?

Ans: The latus rectum of a hyperbola is the line segment perpendicular to the transverse axis and passing through one of the foci. Its length can be calculated using the formula-

$\mathrm{LR}=\frac{2{\mathrm{b}}^2}{\mathrm{a}}$

Q. How do you find the vertices of a hyperbola? 

Ans: For a hyperbola with the equation $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$, the vertices are located at the points (±a, 0). For a hyperbola with the equation  $\frac{y^2}{b^2}-\frac{x^2}{a^2}=1$, vertices are (0, ±b)