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

## Example Questions

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### Example Question #11 : 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 #12 : 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.

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.

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 #13 : 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 vertical and the endpoints are  and .

### Example Question #14 : 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.

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.

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 vertical and the endpoints are  and .

### Example Question #15 : Find The Endpoints Of The Major And Minor Axes Of An Ellipse

Which is not the endpoint of a major or minor axis of the ellipse

Explanation:

The center has an x-coordinate of 4, and the endpoints of the horizontal axis are away from the center. The y-coordinates of these endpoints are the same as the center, -2. So, these points are or and .

The center has a y-coordinate of -2, and the endpoints of the vertical axis are away from the center. The x-coordinates of these endpoints are the same as the center, 4. So, these points are or and .

The only point not listed is , which is , so that's the correct answer choice.

### Example Question #16 : Find The Endpoints Of The Major And Minor Axes Of An Ellipse

The center of an ellipse is and the foci are at  and . If the length of the minor axis is 22, what are the endpoints of the major axis?

and

Explanation:

To figure out the endpoints of the major axis, we need to know its length, which we can determine using the equation where a is half the length of the major axis, b is half the length of the minor axis, and c is the distance from the center to the foci.

First, we can determine b. Right now we know that the full length of the minor axis is 22, so half its length is 11. In other words, .

Now we can determine the distance from the foci to the center. The y-coordinates stay the same, so we will be comparing the x-coordinates. The center is and one of the foci is at - they should both be the same distance away, so we can use either for this part. We can do this more informally, too, but we are solving this equation:

adding 4 gives us , so the distance from the center to the foci is 10, or .

Now we can plug this information into the equation:

Since the foci are on the major axis, and since the distance 10 was added to the x-coordinate, we can conclude that the major axis is horizontal.

This means that our endpoints are

### Example Question #11 : Find The Endpoints Of The Major And Minor Axes Of An Ellipse

The equation of an ellipse is given by

Find the endpoints of the major and minor axes of the ellipse.

Major: (4, 8) and (4, 2)

Minor: (1, 3) and (7, -3)

Major: (-4, -8) and (4, 2)

Minor: (-1, -3) and (7, -3)

Major: (4, -8) and (4, 2)

Minor: (1, 3) and (7, 3)

Major: (4, 8) and (4, 2)

Minor: (1, -3) and (7, 3)

Major: (4, -8) and (4, 2)

Minor: (1, -3) and (7, -3)

Major: (4, -8) and (4, 2)

Minor: (1, -3) and (7, -3)

Explanation:

The major and minor axis run through the center of an ellipse in the vertical and horizontal directions.  Lets take a look at the equation, which is already in standard form

Because the denominator on the y term is larger, the major axis is in the vertical direction.  It's endpoints are located  units away from the center, along the verical axis.  So, the endpoints of the major axis are given by

and

The minor axis is in the horizontal direction and it's endpoints are located  away from the center, along the horizontal axis.  So, the endpoints of the minor axis are given by

and

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