# College Algebra : Hyperbolas

## Example Questions

### Example Question #1 : Hyperbolas

Using the information below, determine the equation of the hyperbola.

Foci:  and

Eccentricity:

Explanation:

General Information for Hyperbola:

Equation for horizontal transverse hyperbola:

Distance between foci =

Distance between vertices =

Eccentricity =

Center: (h, k)

First determine the value of c. Since we know the distance between the two foci is 12, we can set that equal to .

Next, use the eccentricity equation and the value of the eccentricity provided in the question to determine the value of a.

Eccentricity =

Determine the value of

Determine the center point to identify the values of h and k. Since the y coordinate of the foci are 4, the center point will be on the same line. Hence, .

Since center point is equal distance from both foci, and we know that the distance between the foci is 12, we can conclude that

Center point:

Thus, the equation of the hyperbola is:

### Example Question #1 : Hyperbolas

Using the information below, determine the equation of the hyperbola.

Foci:  and

Eccentricity:

Explanation:

General Information for Hyperbola:

Equation for horizontal transverse hyperbola:

Distance between foci =

Distance between vertices =

Eccentricity =

Center: (h, k)

First determine the value of c. Since we know the distance between the two foci is 8, we can set that equal to .

Next, use the eccentricity equation and the value of the eccentricity provided in the question to determine the value of a.

Eccentricity =

Determine the value of

Determine the center point to identify the values of h and k. Since the y coordinate of the foci are 8, the center point will be on the same line. Hence, .

Since center point is equal distance from both foci, and we know that the distance between the foci is 8, we can conclude that

Center point:

Thus, the equation of the hyperbola is:

### Example Question #2 : 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 #1 : 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 .

### Example Question #5 : 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 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 .

First, put the given equation in the standard form of the equation of a hyperbola.

Group the  terms together and the  terms together.

Next, factor out  from the  terms and  from the  terms.

From here, complete the squares. Remember to add the same amount to both sides of the equation!

Subtract  from both sides.

Divide both sides by .

Factor both terms to get the standard form for the equation of a hyperbola.

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 #3 : 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 .

First, put the given equation in the standard form of the equation of a hyperbola.

Group the  terms together and the  terms together.

Next, factor out  from the  terms and  from the  terms.

From here, complete the squares. Remember to add the same amount to both sides of the equation!

Subtract  from both sides.

Divide both sides by .

Factor both terms to get the standard form for the equation of a hyperbola.

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 center and the vertices of the following hyperbola:

Explanation:

In order to find the center and the vertices of the hyperbola given in the problem, we must examine the standard form of a hyperbola:

The point (h,k) gives the center of the hyperbola. We can see that the equation in this problem resembles the second option for standard form above, so right away we can see the center is at:

In the first option, where the x term is in front of the y term, the hyperbola opens left and right. In the second option, where the y term is in front of the x term, the hyperbola opens up and down. In either case, the distance tells how far above and below or to the left and right of the center the vertices of the hyperbola are. Our equation is in the first form, where the x term is first, so the hyperbola opens left and right, which means the vertices are a distance  to the left and right of the center. We can now calculate  by identifying it in our equation, and then go 3 units to the left and right of our center to find the following vertices:

### Example Question #4 : Hyperbolas

Find the equations of the asymptotes for the hyperbola with the following equation:

Explanation:

For a hyperbola with its foci on the -axis, like the one given in the equation, recall the standard form of the equation:

, where  is the center of the hyperbola.

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

Group the  terms together and  terms together.

Factor out  from the  terms and  from the  terms.

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

Add  to both sides of the equation:

Divide both sides by .

Factor the two terms to get the standard form of the equation of a hyperbola.

The slopes of this hyperbola are given by the following:

For the hyperbola in question,  and .

Thus, the slopes for its asymptotes are .

Now, plug in the center of the hyperbola,  into the point-slope form of the equation of a line to get the equations of the asymptotes.

For the first equation,

For the second equation,

### Example Question #5 : Hyperbolas

Which of the following correctly describes the hyperbola of the equation

.

A horizontal hyperbola with center at .

A vertical hyperbola with center at .

A vertical hyperbola with center at .

A horizontal hyperbola with center at .