You ran into trouble when you went from to and . You don't know the actual values of c and a, just the ratio (you could have chosen c=6 and a=8 instead, and would still be 0.75.
Try substituting . Then you can go two ways (and get two solutions). Either substitute or . If you think about it, there should be two solutions, a "tall, skinny" ellipse and a "short, fat" ellipse.
So the first one will be , and the second will be .
I think you can find the solutions from there. If you have any further questions, please post again in this thread.