Your error is in thinking that you are necessarily reducing the 
fractions to lowest terms.  Instead think of it as making the 
numerators 1.  Sometimes it amounts to reducing the fraction
but not always.

Do it this way instead. Go back to:



Get a  coefficient where the  is by 
multiplying top and bottom of the first fraction by
, and get a  coefficient where 
the  is on the second fraction by multiplying 
top and bottom by , and we have this:



Now cancel and that leaves just 1 understood on top:



So we have



And we only need to reduce the fractions on the bottom



The larger denominator is , and since ,

we compare the above to:



So 

            
         


Eccentricity of an ellipse =  where 












Eccentricity of this elipse = 
  
Edwin