A ladder ellipse problem

We consider a problem similar to the well-known ladder box problem, but where the box is replaced by an ellipse. A ladder of a given length, $s$, with ends on the positive x and y axes, is known to touch an ellipse that lies in the first quadrant and is tangent to the positive x and y axes. We then want to find the height of the top of the ladder above the floor. We show that there is a value, $s = s_0$, such that there is only one possible position of the ladder, while if $s > s_0$, then there are two different possible positions of the ladder. Our solution involves solving an equation which is equivalent to solving a 4th degree polynomial equation.