Take two distinct coprime integers $a,b$ and take $q=a^2+b^2$. Consider the equation $(xy^{-1})^2\equiv-1\bmod q$. The solutions are two roots which correspond to $(x,y)=(a,b)$ and $(x,y)=(b,a)$. Look at the representative $\alpha$ of $(xy^{-1})\bmod q$ where $\alpha\in[-q/2,q/2)$ holds.

Is there structure to $q$ that achieve large ratio $r=\frac{|\alpha|}{\sqrt q}$ ($r=2$ is achievable at $b=a-1$)?

If $a,b$ are randomly picked with $a<b<2a$ then what is the probability $r<q^\epsilon$ holds for fixed $\epsilon\in[0,\frac12]$?