# Precalculus : Pre-Calculus

## Example Questions

### Example Question #86 : Understand Features Of Hyperbolas And Ellipses

Find the vertices of a 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 .

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

### Example Question #87 : Understand Features Of Hyperbolas And Ellipses

Find the endpoints of the conjugate 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 endpoints of the conjugate axis are located at .

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

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

### Example Question #88 : Understand Features Of Hyperbolas And Ellipses

Find the endpoints of the conjugate axis for 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 .

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

### Example Question #89 : Understand Features Of 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.

Factor out  from the  terms.

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

Add  to both sides of the equation.

Factor the square portion 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 #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: