Let $k$ be a characteristic zero field and let $L/k$ be a quadratic extension. Write $L = k(\sqrt{p})$.
Let $q$ be a non-square in $k^\star$ and let $r \in k^\star$ be any constant.
Consider the fraction field $F$ of field $k[X,Y]/(X^2 - qY^2 - r)$ (function field of a conic).
Under which conditions on $p$, $q$ and $r$ are the fields $L$ and $F$ linearly disjoint?
(Does there exist a "nice" answer to this question?)
If I'm right (I hope I am?), this question is equivalent to:
when do there exist coprime $F,G \in k[X,Y]$ such that $X^2 - qY^2 - r$ divides $F^2 - pG^2$?
This is a concrete question, but I don't know whether it has a concrete answer.