Since $F^TF$ and $G^TG$ are symmetric positive definite matrices, there is a $2\times2$ nonsingular matrix $A$ such that $C:=A^TF^TFA$ and $D:=A^TG^TGA$ are diagonal positive definite matrices, say with the diagonal entries $c_1,c_2$ and $d_1,d_2$, respectively. So, $k_1=\eta^TC\eta=c_1\eta_1^2+c_2\eta_2^2$ and $k_2=\eta^TD\eta=d_1\eta_1^2+d_2\eta_2^2$, where $[\eta_1,\eta_2]^T=\eta:=A^{-1}\xi$.
Solving now the equations $k_1=c_1\eta_1^2+c_2\eta_2^2$ and $k_2=d_1\eta_1^2+d_2\eta_2^2$ for $\eta_1$ and $\eta_2$, one finds the expressions of $\xi=A\eta$ in terms of $k_1$ and $k_2$.