Condition of two hyperbolas do not intersect Given two hyperbolas h1(with foci and center) and h2(with foci and center), In what condition these hyperbolas will not intersect to each other?. I can get the condition when h1 and h2 are standard hyperbola (parallel to axis and the center is the origin). I want to find the condition when both of them are not standard hyperbola. Thanks

Let us consider the reference equilateral hyperbola with equation $xy=1$ (see blue curve on graphics below). Any rectangular hyperbola can be obtained from the reference with a rotation (angle $\theta$) followed by a translation (vector $\binom{a}{b}$). One should obtain the following result : 


*

*if $\theta\neq0$, there are intersection points. 

*If $\theta=0$, the only translations that give no intersection point are with vector $k\binom{1}{1}$ for $-2<k<2$ 
(see the example of the magenta curve).
I thought I have a proof but it is not the case.

