$x$ and $y$ are two scalar unknowns. The two equations are
$$|\mathbf{I}+x\mathbf{h}_1\mathbf{h}'_1+y\mathbf{h}_2\mathbf{h}'_2|=R$$
and
$$\mathbf{h}'_1\left(\mathbf{I}+x\mathbf{h}_1\mathbf{h}'_1+y\mathbf{h}_2\mathbf{h}'_2\right)^{-1}\mathbf{h}_1=\mathbf{h}'_2\left(\mathbf{I}+x\mathbf{h}_1\mathbf{h}'_1+y\mathbf{h}_2\mathbf{h}'_2\right)^{-1}\mathbf{h}_2$$
where $\mathbf{h}_i$ are given column vectors, $()'$ indicates the hermitian, $||$ indicates the determinant, $R$ is a given positive real number, and $\mathbf{I}$ is the identity matrix of proper dimension.
The equations do not look pretty, but from the physical interpretation of the problem, I know there is a unique non-negative solution and I even know the neighbourhood. So is there any way to solve using some readymade matlab function? I tried fsolve, which is not working. Otherwise please suggest some appropriate numerical method which I can code in matlab.