I have tried to describe this problem as best as I can but I realize that (because I am not a trained mathematician) my description may fall short. Therefore I am prepared to edit this question until it is clear and unambiguous.
Consider an ellipse $A$
$$ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 $$
which is fully embedded in a circle $E$
$$ x^2 + y^2 = R^2 $$
Such that there exists a family of ellipses
$$ \frac{x^2}{(sa + (1 - s)R)^2} + \frac{y^2}{(sb + (1 - s)R)^2} = 1 $$
parametrized by $0 \le s \le 1$.
I'd like to determine the equation of the curve (starting) from an arbitrary point $P(x,y)$ on $A$ to $E$ such that the curve is normal to the family of ellipses, as shown in the figure below.
All I know is that this problem has some characteristics of an initial value ODE.