I think this must be well-known (and probably not hard to prove either), but I cannot find a reference: for a (positive) rational number $a$, the number of integral solutions to the equation $$ x^2+y^2-az^2=0, $$ with $|x|,|y|,|z|<T$ is $C(a) T \log T$, where $C(a)$ is a constant depending only on $a$. I would very much appreciate a reference which also includes a proof.