I'm working in a region delimited by 4 circumferences, concentric 2 to 2, with opposite centers (shaded area) which vary with respect to a parameter $\alpha$.
$$r<(x-a(\alpha))^2+y^2<R$$
$$r<(x+a(\alpha))^2+y^2<R$$
This region is symmetric respect the OY axis. I would like to obtain the largest circle contained in this region. What do you think about it? It's possible?
From your nice visualisation, you can see that the desired circle would be tangent to all four circles. For it to be tangent to a pair of concentric circles, its radius must be $\rho=\frac{R-r}2$, and its centre must be at distance $\frac{R+r}2$ from their two centres. Also the centre must clearly lie on the $y$ axis. Its $y$ coordinate can then be obtained from
$$ a(\alpha)^2+y^2=\left(\frac{R+r}2\right)^2 $$
as
$$ y=\sqrt{\left(\frac{R+r}2\right)^2-a(\alpha)^2}\;. $$