Precalculus : Find the Center and Foci of an Ellipse

Study concepts, example questions & explanations for Precalculus

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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 

Explanation:

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 #41 : Ellipses

Find the center of the ellipse with the following equation:

Possible Answers:

Correct answer:

Explanation:

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 #1 : Find The Center And Foci Of An Ellipse

Find the center of the ellipse with the following equation:

Possible Answers:

Correct answer:

Explanation:

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 #2 : Find The Center And Foci Of An Ellipse

Find the center of the ellipse with the following equation:

Possible Answers:

Correct answer:

Explanation:

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:

Explanation:

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 #4 : Find The Center And Foci Of An Ellipse

Find the center of the ellipse with the following equation:

Possible Answers:

Correct answer:

Explanation:

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 #5 : Find The Center And Foci Of An Ellipse

Find the center of the ellipse with the following equation:

Possible Answers:

Correct answer:

Explanation:

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 #6 : Find The Center And Foci Of An Ellipse

Find the foci of an ellipse with the following equation:

Possible Answers:

Correct answer:

Explanation:

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 #7 : Find The Center And Foci Of An Ellipse

Find the foci of the ellipse with the following equation:

Possible Answers:

Correct answer:

Explanation:

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 #8 : Find The Center And Foci Of An Ellipse

Find the foci for the ellipse with the following equation:

Possible Answers:

Correct answer:

Explanation:

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 .

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