How can we find some radius of circle so $-x\arctan(x)+0.2x-y\arctan(y)+0.9y=0$ will be fully inside this circle?

Question

If he have this region $$ \begin{align} \ -x\arctan(x)+0.2x-y\arctan(y)+0.9y=0\\ \end{align} $$ How can we find some $R$ (maybe minimum) so this region will fully inside this circle $(x-a)^2+(y-b)^2=R^2$.

In WolframAlpha I've found for example $(x-0.2/2)^2+(y-0.9/2)^2=0.8$ so $R=\sqrt0.8$ fits but have no idea how to prove that and how to find minimum radius.

Answer 1

There will be hardly an exact solution to find the minimal radius. If you want to find the maximal and minimal height of the curve, differentiate in respect to $x$, set $y'=0$ and you'll get $$\arctan(x)+\frac{x}{1+x^2}+0.2=0.$$ From here you'll find only a numerical approximation of the solutions.