Alright. The standard form of an ellipse is:
In your equation, we have:
Now, group the x's and y's together:
Factor out a 4 for the x's, and a 9 for the y's:
Now complete the square, for the x's and y's.
Since you are completing the square within the parenthesis, you have to multiply by the coefficient of the parenthesis, and that is how I get 36. ( -64 + 64 + 36 = 36)
Next: Divide by 36
So, now that we have the ellipse in the standard form, we can find the vertices and the foci. The foci of an ellipse =
Where C is the length of the foci.
Since we know that the foci are on the major axis of the ellipse, we can determine the placing of the foci:
is the length of the foci, so now we know that the two foci are at:
The using a and b, we have that the vertices are at:
along the major axis
along the minor axis.
If you had some trouble understanding the terminology, or how I came to find the vertices and the foci, I believe this site will be of help to you:
Focus of Ellipse. The formula for the focus and ...