Precalculus : Hyperbolas and Ellipses

Study concepts, example questions & explanations for Precalculus

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Example Questions

Example Question #21 : Hyperbolas And Ellipses

The equation of an ellipse is given by

Find the center and foci of the ellipse.

Possible Answers:

Center (4, -3), Foci (0, -3) and (8, -3)

Center (4, -3), Foci (-1, -3) and (9, -3)

Center (4, -3), Foci (4, 1) and (4, -7)

Center (4, -3), Foci (4, 2) and (4, -8)

Center (-4, 3), Foci (0, -3) and (8, -3)

Correct answer:

Center (4, -3), Foci (4, 1) and (4, -7)

Explanation:

Begin by noticing the equation is already in standard form

First, the center is given by (h,k).  In this problem, h=4, y=-3 so the center is (4, -3).

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

Therefore, the foci are 4 units from the center.  Because the larger denominator is under the y term, the foci are 4 units in either vertical direction.  Thus, the foci are at 

 AND 

Example Question #22 : Hyperbolas And Ellipses

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 vertical.

Plug in the values to solve for .

The foci are then at the points  and .

Example Question #23 : Hyperbolas And Ellipses

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

Possible Answers:

Correct answer:

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 #1691 : Pre Calculus

Find the eccentricity for the ellipse with the following equation:

Possible Answers:

Correct answer:

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 #25 : Hyperbolas And Ellipses

Find the eccentricity of the ellipse with the following equation:

Possible Answers:

Correct answer:

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 #26 : Hyperbolas And Ellipses

Find the eccentricity of an ellipse with the following equation:

Possible Answers:

Correct answer:

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 #27 : Hyperbolas And Ellipses

Find the eccentricity of the ellipse with the following equation:

Possible Answers:

Correct answer:

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 #28 : Hyperbolas And Ellipses

Find the eccentricity of the ellipse with the following equation:

Possible Answers:

Correct answer:

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 #29 : Hyperbolas And Ellipses

Find the eccentricity of an ellipse with the following equation:

Possible Answers:

Correct answer:

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 #30 : Hyperbolas And Ellipses

Find the eccentricity of the ellipse with the following equation:

Possible Answers:

Correct answer:

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.

 

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