### All Precalculus Resources

## Example Questions

### Example Question #11 : Understand Features Of Hyperbolas And Ellipses

Find the center of the ellipse with the following equation:

**Possible Answers:**

**Correct answer:**

Recall that the standard form of the equation of an ellipse is

, where is the center for the ellipse.

For the equation given in the question, and .

The center of the ellipse is at .

### Example Question #12 : Understand Features Of Hyperbolas And Ellipses

Find the center of the ellipse with the following equation:

**Possible Answers:**

**Correct answer:**

Start by putting the equation back into the standard equation of the ellipse:

, where is the center for the ellipse.

Group the terms and terms together.

Factor out a from the terms, and a from the terms.

Now, complete the square. Remember to add the same amounts on both sides of the equation.

Now, divide both sides by .

Finally, factor the equations to get the standard form of the equation for an ellipse.

Since and , the center for this ellipse is .

### Example Question #13 : Understand Features Of Hyperbolas And Ellipses

Find the center of the ellipse with the following equation:

**Possible Answers:**

**Correct answer:**

Start by putting the equation back into the standard equation of the ellipse:

, where is the center for the ellipse.

Group the terms and terms together.

Factor out a from the terms and a from the terms.

Now, complete the squares. Make sure you add the same amount on both sides!

Subtract from both sides.

Now, divide both sides by .

Finally, factor the terms to get the standard form of the equation of an ellipse.

Since and , the center of the ellipse is .

### Example Question #14 : Understand Features Of Hyperbolas And Ellipses

Find the center of the ellipse with the following equation:

**Possible Answers:**

**Correct answer:**

Start by putting the equation back into the standard equation of the ellipse:

, where is the center for the ellipse.

Group the terms and terms together.

Factor out a from the terms and a from the terms.

Now, complete the squares. Remember to add the same amount on both sides!

Subtract from both sides.

Divide both sides by .

Finally, factor the terms to get the standard form of the equation of an ellipse.

Since and , is the center of this ellipse.

### Example Question #15 : Understand Features Of Hyperbolas And Ellipses

Find the foci of an ellipse with the following equation:

**Possible Answers:**

**Correct answer:**

Recall that the standard form of the equation of an ellipse is

, where is the center for the ellipse.

When , the major axis will lie on the -axis and be horizontal. When , the major axis will lie on the -axis and be vertical.

Recall also that the distance from the center to a focus, , is given by the equation when , and the equation is when .

When the major axis follows the -axis, the points for the foci are and .

When the major axis follows the -axis, the points for the foci are and .

For the given equation, the center is at . Since , the major-axis is vertical.

Plug in the values to solve for .

Now, add to the y-coordinate of the center to get one focus. Subtract from the y-coordinate of the center to get the other focus point.

The foci for the ellipse is then and .

### Example Question #16 : Understand Features Of Hyperbolas And Ellipses

Find the foci of the ellipse with the following equation:

**Possible Answers:**

**Correct answer:**

Recall that the standard form of the equation of an ellipse is

, where is the center for the ellipse.

When , the major axis will lie on the -axis and be horizontal. When , the major axis will lie on the -axis and be vertical.

Recall also that the distance from the center to a focus, , is given by the equation when , and the equation is when .

When the major axis follows the -axis, the points for the foci are and .

When the major axis follows the -axis, the points for the foci are and .

Start by putting the equation into the standard form of the equation of an ellipse.

Group the and terms together.

Now, factor out a from the terms and a 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.

Divide both sides by .

Factor both terms to get the standard form of the equation of an ellipse.

Now, the center for this ellipse is and its major axis is horizontal.

Next, solve for .

The foci for this ellipse are then at and .

### Example Question #17 : Understand Features Of Hyperbolas And Ellipses

Find the foci for the ellipse with the following equation:

**Possible Answers:**

**Correct answer:**

Recall that the standard form of the equation of an ellipse is

, where is the center for the ellipse.

When , the major axis will lie on the -axis and be horizontal. When , the major axis will lie on the -axis and be vertical.

Recall also that the distance from the center to a focus, , is given by the equation when , and the equation is when .

When the major axis follows the -axis, the points for the foci are and .

When the major axis follows the -axis, the points for the foci are and .

For the given equation, the center is at . Since , the major-axis is horizontal.

Plug in the values to solve for .

The foci are then at the points and .

### Example Question #18 : Understand Features Of Hyperbolas And Ellipses

Find the foci of the ellipse with the following equation:

**Possible Answers:**

**Correct answer:**

Recall that the standard form of the equation of an ellipse is

, where is the center for the ellipse.

When the major axis follows the -axis, the points for the foci are and .

When the major axis follows the -axis, the points for the foci are and .

Start by putting the equation into the standard form of the equation of an ellipse.

Group the and terms together.

Factor out a from the terms and a from the terms.

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

Subtract from both sides.

Divide both sides by .

Now, factor both terms to get the standard form of the equation of an ellipse.

The center of the ellipse is . Since , the major axis of this ellipse is horizontal.

Now, find the value of .

The foci of this ellipse are then and .

### Example Question #19 : Understand Features Of Hyperbolas And Ellipses

Find the foci of an ellipse with the following equation:

**Possible Answers:**

**Correct answer:**

Recall that the standard form of the equation of an ellipse is

, where is the center for the ellipse.

When the major axis follows the -axis, the points for the foci are and .

When the major axis follows the -axis, the points for the foci are and .

Start by putting the equation into the standard form of the equation of an ellipse.

Group the and terms together.

Factor out from the terms and a from the terms.

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

Subtract both sides by .

Divide both sides by .

Factor both terms to get the standard form of the equation of an ellipse.

The center of this ellipse is at . Since , the major axis of this ellipse is vertical.

Now, solve for .

The foci for this ellipse are then and .

### Example Question #20 : Understand Features Of Hyperbolas And Ellipses

Find the center and foci of the ellipse

.

**Possible Answers:**

Center: ; Foci:

Center: ; Foci:

Center: ; Foci:

Center: ; Foci:

Center: ; Foci:

**Correct answer:**

Center: ; Foci:

The center of this ellipse is . The number under is bigger than the number under , so the major axis goes up and down. The foci will also be on the major axis, so their x-coordinates will be 0, like the center.

To figure out the distance from the center to the foci, we can use the formula where a is half the major axis, b is half the minor axis, and c is the distance from the center to the foci.

In this case, and :

subtract 36 from both sides

multiply both sides by -1

take the square root

This means that since the center is , the foci are located at and .

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