Let $r\geq 1$ be a real number, $-1\leq x\leq 1$ a real number and $y>2$ a real number. We consider this data to be fixed.
How can I obtain an upper bound on the number of $(a,b,c,d)\in \mathbf{Z}^4$ such that
$\left(c(x^2-y^2) +(d-a)x-b\right)^2+y^2(2cx+d-a)^2 \leq 2(r-1)y^2; $ $ad-bc = 1;$ $ (a,b,c,d) = (1,0,0,1) \mod 2.$
To be clear, the quadruple $(a,b,c,d)$ is supposed to satisfy all three conditions. (I edited the question and added the last two conditions.)
Are there any computer packages that can help me do this?
I expanded the entire expression using Maple, but this didn't look very pleasant.
The motivation lies in the bounding the number of matrices $A$ in $\mathrm{SL}_2(\mathbf{Z})$ such that the geodesic distance between the point $x+iy$ and $A\cdot (x+iy)$ is bounded by $r$.