6-3 Ellipses

6-3 Ellipses
This week I am going to explain how to do ellipses. There are seven steps that are required to solve an ellipse.

The standard form of an ellipse is:

You can do one of two things to put an ellipse in standard form. The first is to divide and the second is to complete the square.
Steps:
1. The first thing that you have to do is to state the major axis. You can find the major axis by looking for the variable with the largest denominator.
2. Next you have to state the minor axis. You can find the minor axis by looking for the variable with the smallest denominator.
3. Then you must find the vertices. You can do this by taking the square root of the largest denominator. This gives you a positive and negative answer. You then put that into point form. If the major is x then the numbers you get go into the x coordinate. If the major is y then the numbers you get go into the y coordinate. You should now have two points.(x,0)(-x,0) or (0,y)(0,-y)
4. Now you have to find the other intercepts. To do this you take the square root of the smallest denominator. Again, you get a positive and negative answer to put in point form. You put these numbers in the coordinate of the minor axis. (x,0)(-x,0) or (0,y)(0,-y). You should now have four points total.
5. To find the length of the major axis you multiply 2 * the square root of the largest denominator. This will give you one number.
6. To find the length of the minor axis you multiply 2* the square root of the smallest denominator. This will also give you one number.
7. Time to find the last piece of the puzzle, the focus. To find it, follow this formula: smallest denominator=largest denominator – focus^2. It will give you a positive and negative number to put in point form. The numbers go in the coordinate of the major axis. (x,0)(-x,0) or (0,y)(0,-y).
After you finish all of the steps you have to sketch the graph.

Example
See numbered steps above for details of each step.
(x^2/4)+(y^2/9)=1
1. Y is the major axis
2. X is the minor axis
3. Square root of 9= +/- 3 (0,3) (0,-3)
4. Square root of 4=+/- 2 (2,0)(-2,0)
5. 2 * square root of 9= 2*3= 6(length of major axis)
6. 2 * square root of 4= 2* 2= 4(length of minor axis)
7. Focus: 4=9-f^2
-f^2=-5
f^2=5
f= +/- 5^1/2 (0, 5^1/2) (0,-5^1/2)
Here is the sketch of the graph: (The red points are the vertices and the green points are the foci)