Example: Find the center, foci, and vertices of the ellipse.  Graph the equation.

Solution:

Finding the center:

There are two standard orientations of ellipse’s, both of them have the center at (h,k).

Horizontal: 

Vertical:

For this example the center of the ellipse is (1,-1)

Finding the orientation:

The larger denominator will indicate the orientation.  Since our example has a larger denominator under the y variable the orientation is vertical.  That means the ellipse is longer vertically.

Finding the vertices: (Major Axis)

a is the distance between the center and the vertex along the major or longer axis.  Since our larger denominator is 64 a is 8.  I know this because the in the formula the larger denominator is . If then .

Since the ellipse is vertically oriented the vertices will be 8 units above and below the center.

This puts the vertices at (1,7) and (1,-9).

Finding the vertices: (Minor Axis)

b is the distance between the center and the vertex along the minor or shorter axis.  Since our smaller denominator is 25 b is 5.  I know this because the in the formula the smaller denominator is . If then .

Since the ellipse is vertically oriented the vertices along the minor axis or shorter axis will be 5 units left and right of the center.

This puts the vertices at (-4,-1) and (6,-1).

Finding the foci: 

c is the distance between the center and the focus.  You can not find c from the equation.  You must know that a, b, and c are related in a similar formula to Pythagorean theorem.

Fill in a and b, then you will be able to solve for c.

Since c is a distance we will use the positive solution.  .

Since the ellipse is vertically oriented the foci will be above and below the center.

This puts the foci at and