## Summary

IDefinition, vocabulary and graphic approachIIProperties of hyperbolasACenter, vertices, and foci of a hyperbolaBLengths of the transverse axisCEccentricityIIIEquation of hyperbolas## Definition, vocabulary and graphic approach

The distance between two points \left(x_0,y_0\right) and \left(x_1,y_1\right) is provided by the distance formula:

\sqrt{\left(x_1-x_0\right)^2+\left(y_1-y_0\right)^2}

The distance between the point \left(2{,}3\right) and the point \left(-2{,}4\right) is:

\sqrt{\left(-2-2\right)^2+\left(4-3\right)^2}=\sqrt{17}

Hyperbolas are often realized as the graphs of inverse functions such as f\left(x\right)=\dfrac{1}{x}. However, not all hyperbolas are graphs of functions.

### Hyperbola

A hyperbola is defined in terms of two foci points A,B and a constant c. A point is on the hyperbola if and only if the difference between the distances from the point on the hyperbola to A and B is c.

## Properties of hyperbolas

### Center, vertices, and foci of a hyperbola

#### Center of a hyperbola

The center *C* of a hyperbola is the midpoint of the foci of the hyperbola.

#### Vertices of a hyperbola

The line segment joining the foci of a hyperbola intersect the hyperbola in two points. These two points, *V* and *W*, are the vertices of the hyperbola.

#### Transverse axis

The transverse axis of a hyperbola is the line segment joining the two vertices of the hyperbola.

The midpoint of the transverse axis of a hyperbola is the center of the hyperbola.

#### Conjugate axis

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

A hyperbola is unchanged when reflected across its conjugate axis.

### Lengths of the transverse axis

#### Length of transverse axis

Suppose the distance from the center of a hyperbola to a vertex of the hyperbola is a. Then the length of the transverse axis is 2a.

Consider the hyperbola in the following graphic:

The distance from the center of the hyperbola to one of its vertices is:

\sqrt{\left(2-1\right)^2+\left(3-0\right)^2}=\sqrt{10}

Therefore, the length of the transverse axis is:

2\sqrt{10}

### Eccentricity

#### Eccentricity

Consider a hyperbola. There is a number e called the eccentricity of the hyperbola such that if P is any point on the hyperbola, a is the distance from the point P to the nearest focus of the hyperbola and b is the distance from the point P to the conjugate axis of the hyperbola, then:

\dfrac{a}{b}=e

The following graphic contains three hyperbolas with different values for eccentricity.

Smaller values of eccentricity correspond to more "curviness" of the hyperbola around its foci.

## Equation of hyperbolas

For the remainder of this section, we will focus on the equations whose graphs are hyperbolas with conjugate axis as either a horizontal or vertical line.

The graph of the function f\left(x\right)=\dfrac{1}{x} is a hyperbola whose conjugate axis is the line y=-x.

### Equation of a hyperbola I

Consider the following equation:

** \dfrac{\left(x-h\right)^2}{a^2}-\dfrac{\left(y-k\right)^2}{b^2}=1 **

The set of points in the xy -plane which satisfy the above equation is a hyperbola with the following properties:

- The center of the hyperbola is \left(h,k\right).
- The foci of the hyperbola are at \left(h+2a,k\right) and \left(h-2a,k\right).
- The vertices of the hyperbola are at \left(h+a,k\right) and \left(h-a,k\right).

The above equation can only be used to describe hyperbolas whose conjugate axis is a vertical line. For example, the hyperbola defined by f\left(x\right)=\dfrac{1}{x} is not covered by this theorem.

Consider the following equation:

\dfrac{\left(x-1\right)^2}{4}-\dfrac{\left(y-2\right)^2}{9}=1

The set of points in the xy -plane is a hyperbola with the following properties:

- The center of the hyperbola is \left(1{,}2\right).
- The foci of the hyperbola are at \left(5{,}2\right) and \left(-3{,}2\right).
- The vertices of the hyperbola are at \left(3{,}2\right) and \left(-1{,}2\right).

### Equation of a hyperbola II

Consider the following equation:

** \dfrac{\left(y-h\right)^2}{a^2}-\dfrac{\left(x-k\right)^2}{b^2}=1 **

The set of points in the xy -plane which satisfy the above equation is a hyperbola with the following properties:

- The center of the hyperbola is \left(k,h\right).
- The foci of the hyperbola are at \left(k,h+2a\right) and \left(k,h-2a\right).
- The vertices of the hyperbola are at \left(k,h+a\right) and \left(k,h-a\right).

The above equation can only be used to describe hyperbolas whose conjugate axis is a horizontal line.

Consider the following equation:

\dfrac{\left(y-3\right)^2}{9}-\dfrac{\left(x-3\right)^2}{9}=1

Then the set of points in the xy -plane is a hyperbola with the following properties:

- The center of the hyperbola is \left(3{,}3\right)
- The foci of the hyperbola are at \left(3{,}7\right) and \left(3,-1\right)
- The vertices of the hyperbola are at \left(3{,}5\right) and \left(3{,}1\right)