# Precalculus : Find the Eccentricity of an Ellipse

## Example Questions

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### Example Question #1 : Find The Eccentricity Of An Ellipse

The equation of an ellipse, , is . Which of the following is the correct eccentricity of this ellipse?

Explanation:

The equation for the eccentricity of an ellipse is , where  is eccentricity,  is the distance from the foci to the center, and  is the square root of the larger of our two denominators.

Our denominators are  and , so .

To find , we must use the equation , where  is the square root of the smaller of our two denominators.

This gives us , so .

Therefore, we can see that

.

### Example Question #2 : Find The Eccentricity Of An Ellipse

Find the eccentricity for the ellipse with the following equation:

Explanation:

Recall that the eccentricity is a measure of the roundness of an ellipse. Use the following formula to find the eccentricity, .

Next, find the distance from the center to the focus of the ellipse, . Recall that when , the major axis will lie along the -axis and be horizontal and that when , the major axis will lie along the -axis and be vertical.

is calculated using the following formula:

for , or

for

For the ellipse in question, , so

Now that we have found the distance from the center to the foci, we need to find the distance from the center to the vertex.

Because , the major axis for this ellipse is horizontal.  will be the distance from the center to the vertices.

For this ellipse, .

Now, plug in the distance from the center to the focus and the distance from the center to the vertex to find the eccentricity of this ellipse.

### Example Question #2 : Find The Eccentricity Of An Ellipse

Find the eccentricity of the ellipse with the following equation:

Explanation:

Recall that the eccentricity is a measure of the roundness of an ellipse. Use the following formula to find the eccentricity, .

Next, find the distance from the center to the focus of the ellipse, . Recall that when , the major axis will lie along the -axis and be horizontal and that when , the major axis will lie along the -axis and be vertical.

is calculated using the following formula:

for , or

for

For the ellipse in question, , so

Now that we have found the distance from the center to the foci, we need to find the distance from the center to the vertex.

Because , the major axis for this ellipse is horizontal.  will be the distance from the center to the vertices.

For this ellipse, .

Now, plug in the distance from the center to the focus and the distance from the center to the vertex to find the eccentricity of this ellipse.

### Example Question #3 : Find The Eccentricity Of An Ellipse

Find the eccentricity of an ellipse with the following equation:

Explanation:

Recall that the eccentricity is a measure of the roundness of an ellipse. Use the following formula to find the eccentricity, .

Next, find the distance from the center to the focus of the ellipse, . Recall that when , the major axis will lie along the -axis and be horizontal and that when , the major axis will lie along the -axis and be vertical.

is calculated using the following formula:

for , or

for

For the ellipse in question,

Now that we have found the distance from the center to the foci, we need to find the distance from the center to the vertex.

Because , the major axis for this ellipse is horizontal.  will be the distance from the center to the vertices.

For this ellipse, .

Now, plug in the distance from the center to the focus and the distance from the center to the vertex to find the eccentricity of this ellipse.

### Example Question #4 : Find The Eccentricity Of An Ellipse

Find the eccentricity of the ellipse with the following equation:

Explanation:

Recall that the eccentricity is a measure of the roundness of an ellipse. Use the following formula to find the eccentricity, .

Next, find the distance from the center to the focus of the ellipse, . Recall that when , the major axis will lie along the -axis and be horizontal and that when , the major axis will lie along the -axis and be vertical.

is calculated using the following formula:

for , or

for

For the ellipse in question,

Now that we have found the distance from the center to the foci, we need to find the distance from the center to the vertex.

Because , the major axis for this ellipse is vertical.  will be the distance from the center to the vertices.

For this ellipse, .

Now, plug in the distance from the center to the focus and the distance from the center to the vertex to find the eccentricity of this ellipse.

### Example Question #6 : Find The Eccentricity Of An Ellipse

Find the eccentricity of the ellipse with the following equation:

Explanation:

Recall that the eccentricity is a measure of the roundness of an ellipse. Use the following formula to find the eccentricity, .

Next, find the distance from the center to the focus of the ellipse, . Recall that when , the major axis will lie along the -axis and be horizontal and that when , the major axis will lie along the -axis and be vertical.

is calculated using the following formula:

for , or

for

For the ellipse in question,

Now that we have found the distance from the center to the foci, we need to find the distance from the center to the vertex.

Because , the major axis for this ellipse is vertical.  will be the distance from the center to the vertices.

For this ellipse, .

Now, plug in the distance from the center to the focus and the distance from the center to the vertex to find the eccentricity of this ellipse.

### Example Question #5 : Find The Eccentricity Of An Ellipse

Find the eccentricity of an ellipse with the following equation:

Explanation:

Recall that the eccentricity is a measure of the roundness of an ellipse. Use the following formula to find the eccentricity, .

Next, find the distance from the center to the focus of the ellipse, . Recall that when , the major axis will lie along the -axis and be horizontal and that when , the major axis will lie along the -axis and be vertical.

is calculated using the following formula:

for , or

for

For the ellipse in question,

Now that we have found the distance from the center to the foci, we need to find the distance from the center to the vertex.

Because , the major axis for this ellipse is horizontal.  will be the distance from the center to the vertices.

For this ellipse, .

Now, plug in the distance from the center to the focus and the distance from the center to the vertex to find the eccentricity of this ellipse.

### Example Question #6 : Find The Eccentricity Of An Ellipse

Find the eccentricity of the ellipse with the following equation:

Explanation:

Start by putting this equation in the standard form of the equation of an ellipse:

, where  is the center of the ellipse.

Group the  terms together and the  terms together.

Factor out a  from the  terms and  from the  terms.

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

Subtract  from both sides.

Divide both sides by .

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

Recall that the eccentricity is a measure of the roundness of an ellipse. Use the following formula to find the eccentricity, .

Next, find the distance from the center to the focus of the ellipse, . Recall that when , the major axis will lie along the -axis and be horizontal and that when , the major axis will lie along the -axis and be vertical.

is calculated using the following formula:

for , or

for

For the ellipse in question,

Now that we have found the distance from the center to the foci, we need to find the distance from the center to the vertex.

Because , the major axis for this ellipse is horizontal.  will be the distance from the center to the vertices.

For this ellipse, .

Now, plug in the distance from the center to the focus and the distance from the center to the vertex to find the eccentricity of this ellipse.

### Example Question #34 : Graphs

Find the eccentricity of an ellipse with the following equation:

Explanation:

Start by putting this equation in the standard form of the equation of an ellipse:

, where  is the center of the ellipse.

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!

Subtract  from both sides.

Divide both sides by .

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

Recall that the eccentricity is a measure of the roundness of an ellipse. Use the following formula to find the eccentricity, .

Next, find the distance from the center to the focus of the ellipse, . Recall that when , the major axis will lie along the -axis and be horizontal and that when , the major axis will lie along the -axis and be vertical.

is calculated using the following formula:

for , or

for

For the ellipse in question,

Now that we have found the distance from the center to the foci, we need to find the distance from the center to the vertex.

Because , the major axis for this ellipse is vertical.  will be the distance from the center to the vertices.

For this ellipse, .

Now, plug in the distance from the center to the focus and the distance from the center to the vertex to find the eccentricity of this ellipse.

### Example Question #7 : Find The Eccentricity Of An Ellipse

Find the eccentricity of the ellipse with the following equation:

Explanation:

Start by putting this equation in the standard form of the equation of an ellipse:

, where  is the center of the ellipse.

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!

Subtract  from both sides.

Divide both sides by .

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

Recall that the eccentricity is a measure of the roundness of an ellipse. Use the following formula to find the eccentricity, .

Next, find the distance from the center to the focus of the ellipse, . Recall that when , the major axis will lie along the -axis and be horizontal and that when , the major axis will lie along the -axis and be vertical.

is calculated using the following formula:

for , or

for

For the ellipse in question,

Now that we have found the distance from the center to the foci, we need to find the distance from the center to the vertex.

Because , the major axis for this ellipse is horizontal.  will be the distance from the center to the vertices.

For this ellipse, .

Now, plug in the distance from the center to the focus and the distance from the center to the vertex to find the eccentricity of this ellipse.

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