# Precalculus : Find the Endpoints of the Major and Minor Axes of an Ellipse

## Example Questions

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### Example Question #1 : 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:

### Example Question #2 : Find The Endpoints Of The Major And Minor Axes Of An Ellipse

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

and

and

and

and

and

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 #1 : Find The Endpoints Of The Major And Minor Axes Of An Ellipse

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

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 #1 : Find The Endpoints Of The Major And Minor Axes Of An Ellipse

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

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 #1 : Find The Endpoints Of The Major And Minor Axes Of An Ellipse

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

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 #6 : Find The Endpoints Of The Major And Minor Axes Of An Ellipse

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

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 #7 : Find The Endpoints Of The Major And Minor Axes Of An Ellipse

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

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!

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 #11 : Ellipses

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

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 #1 : Find The Endpoints Of The Major And Minor Axes Of An Ellipse

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

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 #10 : Find The Endpoints Of The Major And Minor Axes Of An Ellipse

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

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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