The equation $x\_1x\_2+y\_1y\_2=n$ is well-studied (Ingham, Heath-Brown, Deshouillers & Iwaniec, Ismoilov) because it arises in an *additive divisor problem.* The number of solutions in positive integers is $n(c\_2\log^2n+c\_1\log n+c\_0)+O(n^{1-\delta})$ where $c\_i$'s are some explicit arithmetic functions of $n$.


The same tools allow to prove similar formula for the number of solutions of the equation $ax\_1x\_2+by\_1y\_2=n$. In this case $c\_i$'s will depend on $n,a$ and $b$.


Is it possible to find this result in a literature?



The number of solutions of the equation $ax_1x_2+by_1y_2=n$
For a smoothened version of your sum, an asymptotic formula can be found in Duke-Friedlander-Iwaniec: A quadratic divisor problem (Inventiones, 1994), see (1)-(5) there. It is straightforward to "unsmooth" this formula, much like it is done in (6)-(7) of the paper.


