Chapters

## What is Hyperbola?

*"A conic section that is created by an intersection of a right circular cone with a plane at such an angle that both halves of the cones intersect each other is known as a hyperbola"*

In mathematics, a hyperbola is an even curve that lies in a plane. We can identify a hyperbola either by its unique **geometric properties** or by its solutions set of its equations. It has two parts, known as **branches or connected components**. These two parts are like two infinite bows and are **mirror images** of one another.

Remember that we can also define an ellipse as a set of points in a coordinate plane. The same applied to a hyperbola.

*"A set of all points (x,y) in a plane in such a way that it has a positive difference of distances between (x,y) and the foci."*

You may be thinking that this definition of the hyperbola resembles the definition of an ellipse in a coordinate plane, so what is the primary difference between two mathematical figures? Well, we define hyperbola in terms of a difference of two distances, whereas we define ellipse in terms of a sum of two distances.

## Elements of Hyperbola

Like an ellipse, a hyperbola has certain elements as shown in the figure below:

### Axes of Symmetry of Hyperbola

Like ellipse, a hyperbola has two axes of symmetry.

**Transverse axis**: It is a line whose end points are the**vertices of the hyperbola**and passes through the center. The**foci**of the hyperbola are present on the line that has a transverse axis.**Conjugate axis**: It is the line segment whose endpoints are**co-vertices of a hyperbola**and is**perpendicular**(intersect at 90 degrees) to the transverse axis.

### Center

The **midpoint** of both the conjugate and transverse axis is known as the center of the hyperbola. It is the point where both the axes intersect each other.

### Asymptotes

From the center two asymptotes of hyperbola pass. The branches or connected components of the hyperbola approach these asymptotes as the hyperbola retreats from the center.

### Center Rectangle

Its sides pass through each **vertex and co-vertex** of the hyperbola and it is centered at the origin. It is employed to graph a hyperbola and its asymptotes. To draw the asymptotes of the hyperbola, draw and extend the diagonals of the central rectangle.

## Hyperbola Formulas

In this section, we will discuss the hyperbola standard equations and formulas.

## Equations of Hyperbola Centered at the Origin

**Standard equation of a hyperbola with transverse axis parallel to the x-axis**

The transverse axis of a hyperbola can be parallel to the x-axis or the y-axis. The equations of both types of hyperbola vary. A hyperbola with a transverse axis parallel to the x-axis is shown below:

The standard equation of a hyperbola having its transverse axis parallel to the x-axis is given below:

Here:

- The length of the transverse axis is equal to 2a.
- The coordinates of the vertices of this hyperbola are
- The length of the conjugate axis is equal to 2b
- The coordinates of the co-vertices of this type of the hyperbola are
- The distance between the foci is equal to 2c, where c can be calculated by an equation
- The coordinates of the foci are
- The formula of the asymptotes of this type of hyperbola is

**Standard equation of an hyperbola with transverse axis parallel to the y-axis**

A hyperbola with a transverse axis parallel to the y -axis is shown below:

The standard equation of a hyperbola having its transverse axis parallel to the x-axis is given below:

Here:

- The length of the transverse axis is equal to 2a.
- The coordinates of the vertices of this hyperbola are
- The length of the conjugate axis is equal to 2b
- The coordinates of the co-vertices of this type of the hyperbola are
- The distance between the foci is equal to 2c, where c can be calculated by an equation
- The coordinates of the foci are
- The formula for equation of the asymptotes is

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## Equations of Hyperbola Not Centered at the Origin

In this section, we will see what are the equations of the hyperbola that are not centered at the origin.

### Horizontal Transverse axis

The equation of a hyperbola that is not centered at the origin and has its transverse axis parallel to the x-axis is given below:

### Vertical Transverse axis

The equation of a hyperbola that is not centered at the origin and has its transverse axis parallel to the y-axis is given below:

## Eccentricity of the Hyperbola

The formula for finding the eccentricity of a hyperbola is given below:

where and

Now, that we know the formulas of the hyperbolas, let us use the equations and formulas to solve the following examples.

## Example 1

Identify vertices of the hyperbola from the following equation:

### Solution

In the equation , . Hence, .

The vertices of the hyperbola with the horizontal transverse axis are . Hence, the vertices of this equation will be:

and

## Example 2

Identify vertices of the hyperbola from the following equation:

### Solution

In the equation , . Hence, .

The vertices of the hyperbola with the horizontal transverse axis are . Hence, the vertices of this equation will be:

and

The formula for finding asymptotes of hyperbola with the horizontal transverse axis is given below:

The value of b in the equation is and a is equal to . We will substitute these values in the above formula to get the asymptotes of the hyperbola:

Hence, the asymptotes of the hyperbola are and .

## Example 3

Identify the vertices and asymptotes of the hyperbola from the following equation:

### Solution

The equation above shows that the hyperbola has vertical transverse axis because the equation of the hyperbola having a vertical transverse axis is .

The coordinates of the vertices of such a hyperbola are .

In the equation of this example, , hence . Hence, the vertices of the hyperbola will be:

and

To find the asymptotes of the hyperbola, recall the following formula for finding the asymptotes of the hyperbola with vertical transverse axis.

In the equation of this example, and . Substitute these values of a and b in the above formula to get the asymptotes.

Hence, asymptotes of the hyperbola are and .

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