Let's suppose we have two variables $(\alpha, \beta)$ on two functions $k_1, k_2$ that can be defined in terms of matrix relations:

$$ k_1 = \left| \xi^T \mathcal{F}^T \mathcal{F} \xi \right| $$

$$ k_2 = \left| \xi^T \mathcal{G}^T \mathcal{G} \xi \right| $$

where:

$$ \mathcal{F}= \left( \begin{matrix} c_{11} & c_{12} \\ c_{21} & c_{22} \end{matrix} \right) $$

$$ \mathcal{G}= \left( \begin{matrix} d_{11} & d_{12} \\ d_{21} & d_{22} \end{matrix} \right) $$

$$ \xi = \left( \begin{matrix} \alpha \\ \beta \end{matrix} \right) $$

Suppose that the determinants of $\mathcal{F}$ and $\mathcal{G}$ are non-zero. It's possible to invert the relation such that:

$$ \alpha = f(k_1, k_2) $$ $$ \beta = g(k_1, k_2) $$

However the multiple solutions return by the CAS (mathematica) are extremely complicated. Is there a compact way to express the functions $f$ and $g$ supposing we are only interested in positive values for $k_1$ and $k_2$?