Equation of a common tangent to the two hyperbolas $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$and $\frac{y^2}{a^2}-\frac{x^2}{b^2}=1$is

$\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$     $And\frac{y^2}{a^2}-\frac{x^2}{b^2}=1$

$\mathrm{Tan}genttoy=mx\pm \sqrt{a^2{m}^2-b^2}$

If this is also tangent to $\frac{x^2}{{(-b)}^2}-\frac{y^2}{{(-a)}^2}=1$

$a^2{m}^2+b^2=(-b^2)m^2-(-a^2)=a^2-b^2{m}^2$

$(a^2-b^2)m^2=a^2-b^2\Rightarrow m=1$

Hence four common tangent are $y=\pm x\pm \sqrt{a^2-b^2}$