Let $f:\mathbb{R}^2\to\mathbb{R}$ be a rational function, whose numerator and denominator are (second-degree) polynomials of $x$ and $y$.
The problem is to decide whether, for some given $k,k',T\in\mathbb{R}^+$, there exist two values $x_0\in\mathopen[-k,k\mathclose]\subseteq\mathbb{R}$ and $y_0\in\mathopen[-k',k'\mathclose]\subseteq\mathbb{R}$ such that $$T=f(x_0,y_0).$$
How can I solve (if possible) the problem? Is there an algorithm for calculating $x_0,y_0$ whenever they exist?
As stated more formally above, the $x_0$ and $y_0$ we may find, must have an absolute value bounded by $k$ and $k'$ (given constants), respectively. If "plugged into" $f$, they should make $f$ equal the given positive constant $T$.