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
 A: Let $x^2-y^2=1$ be the first hyperbola.


*

*Re-scaling
$$x^2-y^2=c^2 \tag{$c^2 \ne 1$}$$

*Conjugate
$$y^2-x^2=c^2$$

*Translation
\begin{align*}
  (x-h)^2-(y-k)^2 &= 1 \\
  2(ky-hx)+h^2-k^2 &= 0 \\
  x &= \frac{h^2-k^2+2ky}{2h} \\
  (h^2-k^2+2ky)^2-4h^2 y^2 &= 4h^2 \\
  4(h^2-k^2)y^2-4k(h^2-k^2)y+4h^2-(h^2-k^2)^2 &= 0 \\
  \Delta & < 0 \\
  k^2(h^2-k^2)^2-(h^2-k^2)[4h^2-(h^2-k^2)^2] & < 0 \\
  h^2(h^2-k^2)(h^2-k^2-4) &< 0 \\
\end{align*}
$$\fbox{$0<h^2-k^2<4$}$$

A: use the answer of this question to get two equations, set them inequal to each other, then that is your condition, swap them back to focal model.
PS: it might be easier to answer in polar or bipolar coordinates, as they seem more suitable for parabola solving problems
A: 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.

