Question

In a hyperbola portion of tangent intercepted between asymptotes is bisected at the point of contact. Consider a hyperbola whose centre is at origin. A line $x+y=2$ touches this hyperbola at $P(1,1)$ and intersects the asymtotes at $A$ and $B$ such that $AB=6\sqrt{2}$ units.

Angle subtended by $AB$ at centre of the hyperbola is

• A. ${\sin }^{-1}\frac{4}{5}$
• B. ${\sin }^{-1}\frac{2}{5}$
• C. ${\sin }^{-1}\frac{3}{5}$
• D. none of these

Answer 1

The correct option is C ${\sin }^{-1}\frac{3}{5}$

$m_{{}_{OA}}=-\frac{1}{2},m_{{}_{OB}}=-2$
$\tan \theta =\left|\frac{-\frac{1}{2}+2}{1+1}\right|=\frac{3}{4}$
or $\sin \theta =\frac{3}{5}$
or $\theta ={\sin }^{-1}\left(\frac{3}{5}\right)$