Consider the subvariety of Spec $\mathbb{Q}[x,y,z]$ cut out by the equations \begin{eqnarray*} f_1: a_1x^2 - y^2 - b_1^2 & = & 0 \\ f_2 : a_2x^2 - z^2 - b_2^2 & = & 0 \end{eqnarray*} where $a_1,a_2,b_1,b_2$ are positive integers, lets assume all distinct. By Legendre's theorem, the equation $f_i = 0$ has a solution in $\mathbb{Q}$ iff and only if $-1$ is a square modulo $SF(a_i)$, where $SF$ denotes the square-free part of an integer.

Is it possible to give a reasonable description of how the solution set of the system is affected by the properties of $a_1,a_2,b_1,b_2$? Or at least, is it possible to say anything interesting/nontrivial in this direction?