# Precalculus : Hyperbolas and Ellipses

## Example Questions

### Example Question #71 : Understand Features Of Hyperbolas And Ellipses

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 the asymptotes for this hyperbola are given by the following:

For the hyperbola in question,  and .

Thus, the slopes for its asymptotes are .

Now, use the center of the hyperbola, , to plug into the point-slope form of a line to find the equations of the asymptotes.

The first asymptote has the following equation:

The second asymptote has the following equation:

### Example Question #72 : Understand Features Of Hyperbolas And Ellipses

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!

Subtract  from 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 the asymptotes for this hyperbola are given by the following:

For the hyperbola in question,  and .

Thus, the slopes for its asymptotes are .

Plug in the center of the hyperbola into the point-slope form of a line to find the equations of the asymptotes. The center of the hyperbola is .

Now, simplify each equation. For the first equation,

For the second equation,

### Example Question #73 : Understand Features Of Hyperbolas And Ellipses

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.

The slopes of the asymptotes for this hyperbola are given by the following:

For the hyperbola in question,  and .

Thus, the slopes for its asymptotes are

Now, plug in the coordinates for the center of the hyperbola  into the point-slope form of the line to find the equations of the asymptotes.

### Example Question #74 : Understand Features Of Hyperbolas And Ellipses

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.

The slopes of the asymptotes for this hyperbola are given by the following:

For the hyperbola in question,  and .

Thus, the slopes for its asymptotes are .

Now, use the point-slope form of a line in addition to the center of the hyperbola to find the equations of the asymptotes.

The center is at .

The first equation of the asymptote would be the following:

The second equation of the asymptote would be the following:

### Example Question #75 : Understand Features Of Hyperbolas And Ellipses

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.

The slopes of the asymptotes for this hyperbola are given by the following:

For the hyperbola in question,  and .

Since the center is , the equations for its asymptotes are .

### Example Question #76 : Understand Features Of Hyperbolas And Ellipses

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.

The slopes of the asymptotes for 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 a line to find the equations of the asymptotes.

For the first asymptote,

For the second asymptote,

### Example Question #77 : Understand Features Of Hyperbolas And Ellipses

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.

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 fo the line to get the equations of the asymptotes.

The center of the hyperbola is . The equations of the asymptotes are then:

### Example Question #78 : Understand Features Of Hyperbolas And Ellipses

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.

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 an equation of a line to find the equations of the asymptotes. The center of the hyperbola is .

The equations of the asymptotes can be then given by the following:

### Example Question #7 : 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 #79 : Understand Features Of Hyperbolas And Ellipses

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 alien to get the equations for the asymptotes.

The center of the hyperbola is .

The equations for the asymptotes are then: