Find all the affine tangents that are simultaneously tangent to the set $E$ and $H$: $$H=\{(x,y)\in \mathbb R^2:xy=-5\}, E=\{(x,y)\in \mathbb R^2:\frac{x^2}{9}+\frac{y^2}{4}=1\}$$
I know that when the line is tangent to the set A at point a: $T_aA=\{v:<grad(h(a)),v>=0\}$ but I don't know how to combine this condition for both sets $ E, H $ to find all tangents
Any tangent to the given ellipse is $$y=mx \pm \sqrt{9m^2+4}$$, insert this in the Eq. of hyperbola:$xy=-5$, to get $$mx^2+\pm\sqrt{9m^2+4}x+5=0$$ For tangency we impose $B^2=4AC$, Then $$9m^2+4=20m \implies m=2, 2/9$$ The tangents are $$y=2x\pm 2\sqrt{10},~~ y=2x/9\pm2\sqrt{10}/3.$$