In a hyperbola, the portion of the tangent intercepted between the ...

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In a hyperbola, the portion of the tangent intercepted between the asymptotes is bisected at the point of contact.
\( \mathrm{P} \)
Consider a hyperbola whose center is at the origin. A line \( x+y= \)
W 2 touches this hyperbola at \( P(1,1) \) and intersects the asymptotes at \( A \) and \( B \) such that \( A B=6 \sqrt{2} \) units.
The angle subtended by \( A B \) at the center of the hyperbola is
(1) \( \sin ^{-1} \frac{4}{5} \)
(2) \( \sin ^{-1} \frac{2}{5} \)
(3) \( \sin ^{-1} \frac{3}{5} \)
(4) none of these
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