**Algebra for the Ellipse**

The algebra
for the ellipse is somewhat detailed, and since this is just an exposition into
the deeper parts of the ellipse I will demonstrate the algebra for you. The
students will not be responsible for these calculations unless we decide to go
on and do this same material for the hyperbola and the parabola. We start with
this picture to give us a reference for our calculations. Note: This
demonstration was done by me (David Drew) in my EMAT 6550 class with Dr. Andrew
Izsak.

The straight
line to (x,y) is given by _{} and the dashed line is
given by _{}. And when you add both of these up you will get 2a. So
without any more explanation let’s try and discover the algebra for the
ellipse. Remember that were looking for an equation that is similar to _{}. We start with…

_{}+_{}= 2a

_{}= 2a - _{}

_{}

_{}

_{}

_{}

_{}

_{}

_{}

_{}

_{}

_{}

_{}

_{}

And if we let _{}then we’ll have our desired equations of _{}.

From this point on we’ll shy away from
the algebra of the ellipse for two reasons:

1. This is not meant to be an algebra
course and the algebra is so detailed that it gets in the way most of the time

2. The algebra is messy and if we try
to draw an ellipse with algebra it is hard to be concise.