Precalculus : Understand features of hyperbolas and ellipses

Example Questions

Example Question #11 : Understand Features Of Hyperbolas And Ellipses

Find the center of the ellipse with the following equation:

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 #12 : Understand Features Of Hyperbolas And Ellipses

Find the center of the ellipse with the following equation:

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 #13 : Understand Features Of Hyperbolas And Ellipses

Find the center of the ellipse with the following equation:

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 #14 : Understand Features Of Hyperbolas And Ellipses

Find the center of the ellipse with the following equation:

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 #15 : Understand Features Of Hyperbolas And Ellipses

Find the foci of an ellipse with the following equation:

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 #16 : Understand Features Of Hyperbolas And Ellipses

Find the foci of the ellipse with the following equation:

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 #17 : Understand Features Of Hyperbolas And Ellipses

Find the foci for the ellipse with the following equation:

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 .

Example Question #18 : Understand Features Of Hyperbolas And Ellipses

Find the foci of the ellipse with the following equation:

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.

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:

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.

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

.

Center: ; Foci:

Center: ; Foci:

Center: ; Foci:

Center: ; Foci:

Center: ; Foci:

Center: ; Foci:

Explanation:

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 .