# Precalculus : Hyperbolas and Ellipses

## Example Questions

### Example Question #1 : Ellipses

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 #31 : Understand Features Of Hyperbolas And Ellipses

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.

### Example Question #32 : Understand Features Of Hyperbolas And Ellipses

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!

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 #33 : Understand Features Of Hyperbolas And Ellipses

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 #34 : Understand Features Of Hyperbolas And Ellipses

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 #35 : Understand Features Of Hyperbolas And Ellipses

Find the eccentricity of the ellipse

.

Explanation:

To find the eccentrictity, first we need to find c, the distance from the center to the foci. We can use the equation where a and b are the lengths of half the minor and major axes, and c is the distance from the center to the foci.

The eccentricity is where c is half the length of the major axis. In this case, because 49 is greater than 25.

The eccentricity is .

### Example Question #36 : Understand Features Of Hyperbolas And Ellipses

Find the eccentricity of

Explanation:

To find the eccentricity, first find "c" as we would if we were finding the focus. The relationship between a, the radius of the major axis, b, the radius of the minor axis, and c for an ellipse is:

add c squared to both sides

subtract 4 from both sides

take the square root

Since , .

The eccentricity is so here

### Example Question #37 : Understand Features Of Hyperbolas And Ellipses

Find the eccentricity of

Explanation:

First find "c" by using the relationship where a is the radius of the major axis and b is the radius of the minor axis.

add c squared to both sides

subtract 2 from both sides

take the square root of both sides

The eccentricity is . Since

### Example Question #38 : Understand Features Of Hyperbolas And Ellipses

The equation of an ellipse is given by

Find the eccentricity of the ellipse.

Explanation:

The equation for the eccentricity of an ellipse is given by

where c is the distance from the center to the foci and a is the square root of the larger denominator.  To find c, use the equation , where  is the larger denominator and  is the smaller denominator.  Plugging in the values, we have

Plugging the values into the equation gives

### Example Question #39 : Understand Features Of Hyperbolas And Ellipses

Find the endpoints of the major and minor axes of the ellipse described by the following equation:

Explanation:

In order to find the endpoints of the major and minor axes of our ellipse, we must first remember what each part of the equation in standard form means:

The point given by (h,k) is the center of our ellipse, so we know the center of the ellipse in the problem is (8,-2), and we know that the end points of our major and minor axes will line up with the center either in the x or y direction, depending on the axis. The parts of the equation that will tell us the distance from the center to the endpoints of each axis are and . If we take the square root of each, a will give us the distance from the center to the endpoints in the positive and negative x direction, and b will give us the distance from the center to the endpoints in the positive and negative y direction:

Now it is important to consider the definition of major and minor axes. The major axis of an ellipse is the longer one, will the minor axis is the shorter one. We can see that b=5, which means the axis is longer in the y direction, so this is the major axis. To find the endpoints of the major axis, we'll go 5 units from the center in the positive and negative y direction, respectively, giving us:

Similarly, to find the endpoints of the minor axis, we'll go 2 units from the center in the positive and negative x direction, respectively, giving us: