### All Precalculus Resources

## Example Questions

### Example Question #1 : Find The Center And Foci Of An Ellipse

The equation of an ellipse, , is . Which of the following is the correct center and foci of this ellipse?

**Possible Answers:**

Center=, Foci= and

Center=, Foci= and

Center=, Foci= and

Center=, Foci= and

Center=, Foci= and

**Correct answer:**

Center=, Foci= and

Because our equation is already in the format

,

we do not have to manipulate the equation. The center of any ellipse in this form will always be . So in this case, our center will be . To find the foci of the ellipse, we must use the equation , where is the greater of the two denominators in our equation ( and ), is the lesser and is the distance from the center to the foci.

We know that and .

By using , we see that , so .

We now know that the two foci are going to be units in either direction of the center along the greater axis.

Because the greater denominator is under our term containing , our ellipse will have its greater axis going vertically, rather than horizontally. Therefore, our foci will be units above and below our center, at and .

### Example Question #2 : Find The Center And Foci Of An Ellipse

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 #3 : Find The Center And Foci Of An Ellipse

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 #4 : Find The Center And Foci Of An Ellipse

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 #5 : Find The Center And Foci Of An Ellipse

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 #6 : Find The Center And Foci Of An Ellipse

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 #7 : Find The Center And Foci Of An Ellipse

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 #8 : Find The Center And Foci Of An Ellipse

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 #9 : Find The Center And Foci Of An Ellipse

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 #10 : Find The Center And Foci Of An Ellipse

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 .