For Hermitian matrices $A,B \in \mathbb{C}^{n \times n}$, can one readily compute a set of cones that separate the maxima of $$\frac{x'Ax}{x'Bx}$$ among $x$ with unit-norm components?

i.e. where do cubics $x\mapsto(x'Bx)Ax$ and $x\mapsto (x'Ax)Bx$ intersect?

I get the sense that a more general question than mine is solved.