Consider a hyperbola H whose centre is at origin and the line $x+y=2$ touches it at the point $(1,1).$The tangent line $x+y=2$  intersect asymptotes at A and B such that $AB=6\sqrt{2}$ then which of the following is true.

(1,1) is midpoint of AB
A,B= $(1\pm 3\sqrt{2}\left(\frac{-1}{\sqrt{2}}\right),1\pm 3\sqrt{2}\left(\frac{1}{\sqrt{2}}\right))$
      =(-2,4),(4,-2)
Equations of asymptotes are y=-2x, $y=-\frac{1}{2}x$
 (2x+y)(x+2y)=0
Equations of H is (2x+y) (x+2y)=9
Angle between asymptotes is $Ta{n}^{-1}\left(\frac{2\sqrt{h^2-ab}}{a+b}\right)$ 
                                                      $=Ta{n}^{-1}\left(\frac{2\sqrt{\frac{25}{4}-4}}{4}\right)$
                                                       $=Ta{n}^{-1}\left(\frac{3}{4}\right)$