Ellipses are oval shaped designs that look like 2 parabolas stuck together facing one another. Because it is like 2 parabolas, it will have 2 vertexes and 2 foci. Here are 2 different kinds of ellipses.

 

 

 

Vertex

 

Vertex

 

 

 

Focus

Focus

 

 

 

 

Vertex

 

 

Focus

 

 

 

 

 

 

Focus

 

Vertex

 

 

 

 

From the previous pictures the ellipse on the left seems to be running left and right, refer to this by saying that the major axis is horizontal or “with the x.” The ellipse on the right is going up and down, therefore the major axis is “with the y.” The vertexes will always lie on the major axis. Looking again at the picture on the left, if the major axis is left and right then the minor axis runs up and down. Just the opposite can be said of the picture on the right. Here is the general form of the equation of an ellipse.

 

 

To see if the major axis runs left and right or up and down, look at the values of a and b. If a² is larger than b², then it runs left and right. If b² is larger than a², then it runs up and down. There is another value that one must consider and that is the value of c. c is the distance from the center of the ellipse to the foci.

 

 

 

What this equation says is, look at a² and b², if a² is larger put it in the big# spot and if b² is larger put it in the big# spot.

 

Here are two graphs of the two different possibilities.

 

a² is larger

 

 

(0,b)

(a,0)

(-a,0)

 

 

 

(-c,0)

(c,0)

(0,-b)

 

 

b² is larger

 

 

 

 

 

 

(0, a)

(0,c)

(b,0)

(-b,0)

 

(0,-c)

(0,-a)

 

 

 

 

 

If a > b then the major axis goes left and right. If b > a then the major axis will go up and down. It is important to notice that all of these ellipses have a center at (0,0).

 

Example

 

Graph 9x² + 4y² = 36

 

First divide every term by 9 since it is the first number in front of either x² or y².

 

 

Divide every term by 4 to get the y² by itself.

 

 

Now the ellipse is in the proper form to find the variables needed.

 

a² = 4 therefore a = 2

 

b² = 9 therefore a = 3

 

Since b > a, the ellipse opens up and down. Now find c. Use the formula for c.

 

c² = 9 - 4

 

The 9 is the bigger number and the 4 is the smaller one.

 

 

By subtracting the 9 and 4 and then taking the square root.

 

Now one has all the variables required to graph the ellipse. It opens up and down because b is bigger and that means go up and down b(3) units and back and forth a(2) units. The foci are found at units along the major axis.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Answer!

 

 

 

 

 

Example

Graph the ellipse

 

Divide everything by 4.

 

 

Divide everything by 25. This will get the right side to a 1.

 

Now the parabola is in the proper form. It is easy to see that the a value (100) is larger than the b value (25).

 

Next, find the value of c so use the formula .

 

 

Simplify.

 

 

Take the square root of both sides.

 

 

 

To graph this, first notice that the a value is larger so the ellipse goes left and right 10 units. It also goes up and down 5 units. The foci are going to be located on the horizontal axis, which means they are at and .

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Answer!

 

 

 

 

 

Example

Graph the ellipse that has a major axis of length 10 and foci of (0, 6) and (0,-6). Also, write the equation of the ellipse in standard form.

First, graph the foci.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Since the foci lie on the major axis, it is evident that the major axis runs up and down. This means that the b value is larger and thus b² is also. Since the major axis is length 10 that makes the vertices on the major axis at (0, 10) and (0, -10). This also makes it obvious that the b value is 10. Since the foci are 6 units from the center that would make the c value 6. Write all this in the equation for c².

 

6² = 10 ² – a²

 

Simplify.

36 = 100 - a²

 

Solve for a.

-64 = - a²

 

a = 8

 

The a value is the to go back and forth. Graphing is next.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

This is the graph, next is to write the equation in standard from. The a value is 8, making a² = 64, b² is 100. Plug these into the standard form.

 

 

Answer!