# Precalculus : Hyperbolas and Ellipses

## Example Questions

### Example Question #91 : Hyperbolas And Ellipses

Find the center of the hyperbola with the following equation:

Explanation:

Recall the standard form of the equation of a hyperbola can come in two forms:

and

In both cases, the center of the hyperbola is located at .

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

Group the  terms together and the  terms together.

Factor out  from the  terms and  from the  terms.

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

Subtract  from both sides of the equation.

Factor the square portions of the equation.

Divide both sides by  to get the standard form of the equation of a hyperbola.

For the hyperbola in question,  and , so the center is at .

### Example Question #92 : Hyperbolas And Ellipses

Find the vertices of the hyperbola with the following equation:

Explanation:

Recall the standard form of the equation of a hyperbola can come in two forms:

and

In both cases, the center of the hyperbola is located at .

For the hyperbola with the equation , the vertices are located at .

For the hyperbola with the equation , the vertices are located at .

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

Group the  terms together and the  terms together.

Factor out  from the  terms and  from the  terms.

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

Add  to both sides of the equation.

Factor the square portions of the equation.

Divide both sides by  to get the standard form for the equation of a hyperbola.

For the hyperbola in question, the center is located at  and . The vertices must be at  and .

### Example Question #93 : Hyperbolas And Ellipses

Find the vertices of the hyperbola with the following equation:

Explanation:

Recall the standard form of the equation of a hyperbola can come in two forms:

and

In both cases, the center of the hyperbola is located at .

For the hyperbola with the equation , the vertices are located at .

For the hyperbola with the equation , the vertices are located at .

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

Group the  terms together and the  terms together.

Factor out  from the  terms and  from the  terms.

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

Add  to both sides of the equation.

Factor the square portions of the equation.

Divide both sides by  to get the standard form for the equation of a hyperbola.

For the hyperbola in question, the center is located at  and . The vertices must be at  and .

### Example Question #94 : Hyperbolas And Ellipses

Find the endpoints of the conjugate axis of the hyperbola with the following equation:

Explanation:

Recall the standard form of the equation of a hyperbola can come in two forms:

and

In both cases, the center of the hyperbola is located at .

For the hyperbola with the equation , the endpoints of the conjugate axis are located at .

For the hyperbola with the equation , the endpoints of the conjugate axis are located at .

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

Group the  terms together and the  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!

Add  to both sides of the equation.

Factor the squares.

Divide both sides by  to get the standard form for the equation of a hyperbola.

For the hyperbola in question, the center is located at  and . The endpoints of the conjugate axis must be at  and .

### Example Question #95 : Hyperbolas And Ellipses

Find the endpoints of the conjugate axis of the hyperbola with the following equation:

Explanation:

Recall the standard form of the equation of a hyperbola can come in two forms:

and

In both cases, the center of the hyperbola is located at .

For the hyperbola with the equation , the endpoints of the conjugate axis are located at .

For the hyperbola with the equation , the endpoints of the conjugate axis are located at .

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

Group the  terms together and the  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!

Subtract  from both sides of the equation.

Factor the squares.

Divide both sides by  to get the standard form for the equation of a hyperbola.

For the hyperbola in question, the center is located at  and . The endpoints of the conjugate axis must be at  and .

### Example Question #96 : Hyperbolas And Ellipses

Find the eccentricity of the following hyperbola:

Explanation:

In order to find the eccentricity of , first determine the values of  and  from the standard form of the hyperbola:

Use the following formula to calculate eccentricity.  The eccentricity of a hyperbola should always be greater than 1.

Substitute and solve for eccentricity.

### Example Question #97 : Hyperbolas And Ellipses

Find the eccentricity of the hyperbola with the following equation:

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 #98 : Hyperbolas And Ellipses

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 #99 : Hyperbolas And Ellipses

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 #100 : Hyperbolas And Ellipses

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,