Your homework assignment is to draw the ellipse . Where will the foci of your graph be located?
Ellipses Centered at the Origin
The third conic section is an ellipse. Recall that a circle is when a plane sliced through a cone and that plane is parallel to the base of the cone. An ellipse is formed when that plane is not parallel to the base. Therefore, a circle is actually a more specific version of an ellipse.
By definition, an ellipse is the set of all points such that the sum of the distances from two fixed points, called foci (the plural of focus), is constant.
Drawing an Ellipse
We will use the definition of an ellipse to draw an ellipse.
Step 1: On a piece of graph paper, draw a set of axes and plot and . These will be the foci.
Step 2: From the definition, we can conclude a point is on an ellipse if the sum of the distances is always constant. In the picture, and .
Step 3: Determine how far apart the foci are. Then, find and .
Step 4: Determine if the point is on the ellipse.
In this concept, the center of an ellipse will be . An ellipse can have either a vertical or horizontal orientation (see below). There are always two foci and they are on the major axis. The major axis is the longer of the two axes that pass through the center of an ellipse. Also on the major axis are the vertices, which its endpoints and are the furthest two points away from each other on an ellipse. The shorter axis that passes through the center is called the minor axis, with endpoints called covertices. The midpoint of both axes is the center.
HORIZONTAL major axis is the axis with length . minor axis is the axis with length . 


VERTICAL major axis is the axis with length . minor axis is the axis with length . 
Other Important Facts
 is ALWAYS greater than . If they are equal, we have a circle.
 The foci, vertices, and covertices relate through a version of the Pythagorean Theorem:
Let's find the vertices, covertices, and foci of . Then, let's graph the ellipse.
First, we need to determine if this is a horizontal or vertical ellipse. Because , we know that the ellipse will be horizontal. Therefore, making and , making . Using the pictures above, the vertices will be and and the covertices will be and .
To find the foci, we need to use the equation and solve for .
The foci are and .
To graph the ellipse, plot the vertices and covertices and connect the four points to make the closed curve.
Now, let's graph and identify the foci.
This equation is not in standard form. To rewrite it in standard form, the right side of the equation must be 1. Divide everything by 441.
Now, we can see that this is a vertical ellipse, where and .
To find the foci, use .
The foci are and .
Finally, let's write equations for the ellipses with the given characteristics below and centered at the origin.
In either part, you may wish to draw the ellipse to help with the orientation.
 vertex: , covertex:
We can conclude that and . The ellipse is horizontal, because the larger value, , is the value of the vertex. The equation is .
 vertex: , focus:
We know that and and that the ellipse is vertical. Solve for using
The equation is
Examples
Example 1
Earlier, you were asked to determine where the foci of your graph would be located.
This equation is not in standard form. To rewrite it in standard form, the right side of the equation must be 1. Divide everything by 144.
Now, we can see that this is a vertical ellipse, where and .
To find the foci, use .
The foci are therefore and .
Example 2
Find the vertices, covertices, and foci of . Then, graph the equation.
Because the larger number is under , the ellipse is vertical. Therefore, and . Use to find .
vertices: and
covertices: and
foci: and
Example 3
Graph and find the foci.
Rewrite in standard form.
This ellipse is horizontal with and . Find .
The foci are and .
Example 4
Find the equation of the ellipse with covertex , focus and centered at the origin.
Because the covertex is and the ellipse is horizontal. From the foci, we know that . Find .
Review
Find the vertices, covertices, and foci of each ellipse below. Then, graph.
Find the equation of the ellipse, centered at the origin, with the given information.
 vertex: covertex:
 covertex: major axis: 18 units
 vertex: minor axis: 4 units
 vertex: covertex:
 covertex: focus:
 vertex: focus:
 covertex: focus:
 focus: minor axis: 16 units
 Real Life Application A portion of the backyard of the White House is called The Ellipse. The major axis is 1058 feet and the minor axis is 903 feet. Find the equation of the horizontal ellipse, assuming it is centered at the origin.
Answers for Review Problems
To see the Review answers, open this PDF file and look for section 10.5.