# Precalculus : Hyperbolas and Ellipses

## Example Questions

### Example Question #191 : Conic Sections

Find the eccentricity of the following hyperbola:

Explanation:

Recall the standard form of the equation of a horizontal hyperbola:

, where  is the center

For a horizontal hyperbola, use the following equation to find the eccentricity:

, where

For the given hyperbola,

and

Thus,

### Example Question #191 : Conic Sections

Find the eccentricty of the following hyperbola:

Explanation:

Recall the standard form of the equation of a horizontal hyperbola:

, where  is the center

Start by putting the given equation into the standard form of the equation of a horizontal hyperbola.

Group the  terms and  terms together.

Factor out  from the  terms and  from the  terms.

Complete the squares. Remember to add the same number on both sides!

Divide both sides by .

Factor both terms to get the standard equation.

For a horizontal hyperbola, use the following equation to find the eccentricity:

, where

For the given hyperbola,

and

Thus,

### Example Question #103 : Hyperbolas And Ellipses

Find the eccentricity of the following hyperbola:

Explanation:

Recall the standard form of the equation of a vertical hyperbola:

, where  is the center

For a horizontal hyperbola, use the following equation to find the eccentricity:

, where

For the given hyperbola,

and

Thus,

### Example Question #104 : Hyperbolas And Ellipses

Find the eccentricity of the following hyperbola:

Explanation:

Recall the standard form of the equation of a vertical hyperbola:

, where  is the center

For a horizontal hyperbola, use the following equation to find the eccentricity:

, where

For the given hyperbola,

and

Thus,

### Example Question #105 : Hyperbolas And Ellipses

Find the eccentricity of the following hyperbola:

Explanation:

Recall the standard form of the equation of a vertical hyperbola:

, where  is the center

For a horizontal hyperbola, use the following equation to find the eccentricity:

, where

For the given hyperbola,

and

Thus,

### Example Question #106 : Hyperbolas And Ellipses

Find the eccentricity for the following hyperbola:

Explanation:

Recall the standard form of the equation of a vertical hyperbola:

, where  is the center

For a horizontal hyperbola, use the following equation to find the eccentricity:

, where

For the given hyperbola,

and

Thus,

### Example Question #107 : Hyperbolas And Ellipses

Find the eccentricity for the following hyperbola:

Explanation:

Recall the standard form of the equation of a vertical hyperbola:

, where  is the center

For a horizontal hyperbola, use the following equation to find the eccentricity:

, where

For the given hyperbola,

and

Thus,

### Example Question #108 : Hyperbolas And Ellipses

Find the eccentricity of the following hyperbola:

Explanation:

Recall the standard form of the equation of a vertical hyperbola:

, where  is the center

We need to put the given equation into the standard form of the equation.

Start by grouping like terms together.

Factor out  from the  terms and  from the  terms.

Complete the squares. Remember to add the same amount to both sides of the equation!

Divide both sides by .

Factor both terms to get the standard equation.

For a horizontal hyperbola, use the following equation to find the eccentricity:

, where

For the given hyperbola,

and

Thus,

### Example Question #1 : Hyperbolas

Find the foci of the hyperbola with the following equation:

Explanation:

Recall that the standard formula of a hyperbola can come in two forms:

and

, where the centers of both hyperbolas are .

When the term with  is first, that means the foci will lie on a horizontal transverse axis.

When the term with  is first, that means the foci will lie on a vertical transverse axis.

To find the foci, we use the following:

For a hyperbola with a horizontal transverse access, the foci will be located at  and .

For a hyperbola with a vertical transverse access, the foci will be located at  and .

For the given hypebola in the question, the transverse axis is horizontal and its center is located at .

Next, find .

The foci are then located at  and .

### Example Question #2 : Hyperbolas

Find the foci of a hyperbola with the following equation:

Explanation:

Recall that the standard formula of a hyperbola can come in two forms:

and

, where the centers of both hyperbolas are .

When the term with  is first, that means the foci will lie on a horizontal transverse axis.

When the term with  is first, that means the foci will lie on a vertical transverse axis.

To find the foci, we use the following:

For a hyperbola with a horizontal transverse access, the foci will be located at  and .

For a hyperbola with a vertical transverse access, the foci will be located at  and .

For the given hypebola in the question, the transverse axis is horizontal and its center is located at .

Next, find .

The foci are then located at  and .