Precalculus : Hyperbolas and Ellipses

Study concepts, example questions & explanations for Precalculus

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Example Questions

Example Question #41 : Hyperbolas And Ellipses

The equation of an ellipse, , is . Which of the following are the correct end points of the MAJOR axis of this ellipse?

Possible Answers:

 and 

 and 

 and 

 and 

 and 

Correct answer:

 and 

Explanation:

First, we must determine if the major axis is a vertical axis or a horizontal axis. We look at our denominators,  and , and see that the larger one is under the -term. Therefore, we know that the greater axis will be a vertical one.

To find out how far the end point are from the center, we simply take . So we know the end points will be  units above and below our center. To find the center, we must remember that for ,

the center will be .

So for our equation, the center will be  units above and below the center give us  and .

Example Question #41 : Hyperbolas And Ellipses

Find the endpoints of the major axis for the ellipse with the following equation:

Possible Answers:

Correct answer:

Explanation:

Recall the standard form of the equation of an ellipse:

, where  is the center of the ellipse.

When , the major axis is horizontal. In this case,  and  are the endpoints of the major axis.

When  and  are the endpoints of the major axis.

 

For the ellipse in question,  is the center. In addition,  and . Since , the major axis is horizontal and the endpoints are  and 

Example Question #43 : Hyperbolas And Ellipses

Find the endpoints of the major axis for the ellipse with the following equation:

Possible Answers:

Correct answer:

Explanation:

Recall the standard form of the equation of an ellipse:

, where  is the center of the ellipse.

When , the major axis is horizontal. In this case,  and  are the endpoints of the major axis.

When  and  are the endpoints of the major axis.

 

For the ellipse in question,  is the center. In addition,  and . Since , the major axis is vertical and the endpoints are  and .

Example Question #41 : Understand Features Of Hyperbolas And Ellipses

Find the endpoints of the major axis of the ellipse with the following equation:

Possible Answers:

Correct answer:

Explanation:

Recall the standard form of the equation of an ellipse:

, where  is the center of the ellipse.

When , the major axis is horizontal. In this case,  and  are the endpoints of the major axis.

When  and  are the endpoints of the major axis.

 

For the ellipse in question,  is the center. In addition,  and . Since , the major axis is vertical and the endpoints are  and .

Example Question #45 : Hyperbolas And Ellipses

Find the endpoints of the major axis of the ellipse with the following equation:

Possible Answers:

Correct answer:

Explanation:

Recall the standard form of the equation of an ellipse:

, where  is the center of the ellipse.

When , the major axis is horizontal. In this case,  and  are the endpoints of the major axis.

When  and  are the endpoints of the major axis.

 

For the ellipse in question,  is the center. In addition,  and . Since , the major axis is horizontal and the endpoints are  and 

Example Question #46 : Hyperbolas And Ellipses

Find the endpoints of the major axis of the ellipse with the following equation:

Possible Answers:

Correct answer:

Explanation:

Recall the standard form of the equation of an ellipse:

, where  is the center of the ellipse.

Start by putting the equation in the standard form as shown above.

Group the  terms and  terms together.

Factor out  from the  terms and  from the  terms.

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

Add  to both sides.

Divide by  on both sides.

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

Recall that when , the major axis is horizontal. In this case,  and  are the endpoints of the major axis.

When  and  are the endpoints of the major axis.

For the ellipse in question,  is the center. In addition,  and . Since , the major axis is horizontal and the endpoints are  and .

Example Question #47 : Hyperbolas And Ellipses

Find the endpoints of the major axis of the ellipse with the following equation:

Possible Answers:

Correct answer:

Explanation:

Recall the standard form of the equation of an ellipse:

, where  is the center of the ellipse.

Start by putting the equation in the standard form as shown above.

Group the  terms and  terms together.

Factor out  from the  terms and  from the  terms.

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

Subtract  from both sides.

Divide by  on both sides.

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

Recall that when , the major axis is horizontal. In this case,  and  are the endpoints of the major axis.

When  and  are the endpoints of the major axis.

For the ellipse in question,  is the center. In addition,  and . Since , the major axis is horizontal and the endpoints are  and .

Example Question #48 : Hyperbolas And Ellipses

Find the endpoints of the minor axis of the ellipse with the following equation:

Possible Answers:

Correct answer:

Explanation:

Recall the standard form of the equation of an ellipse:

, where  is the center of the ellipse.

When , the minor axis is horizontal. In this case,  and  are the endpoints of the minor axis.

When  and  are the endpoints of the vertical minor axis.

 

For the ellipse in question,  is the center. In addition,  and . Since , the minor axis is horizontal and the endpoints are  and 

Example Question #49 : Hyperbolas And Ellipses

Find the endpoints of the minor axis of the ellipse with the following equation:

Possible Answers:

Correct answer:

Explanation:

Recall the standard form of the equation of an ellipse:

, where  is the center of the ellipse.

When , the minor axis is horizontal. In this case,  and  are the endpoints of the minor axis.

When  and  are the endpoints of the vertical minor axis.

 

For the ellipse in question,  is the center. In addition,  and . Since , the minor axis is horizontal and the endpoints are  and 

Example Question #50 : Hyperbolas And Ellipses

Find the endpoints of the minor axis of the ellipse with the following equation:

Possible Answers:

Correct answer:

Explanation:

Recall the standard form of the equation of an ellipse:

, where  is the center of the ellipse.

When , the minor axis is horizontal. In this case,  and  are the endpoints of the minor axis.

When  and  are the endpoints of the vertical minor axis.

 

For the ellipse in question,  is the center. In addition,  and . Since , the minor axis is horizontal and the endpoints are  and .

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