Question

The range of values of $a$ such that the angle $\theta$ between the pair of tangents drawn from $(a,0)$ to the circle $x^2+y^2=1$ satisfies $\frac{\pi }{2}<\theta <\pi$, lies in

• A. $(-1,1)$
• B. $(1,\sqrt{2})$
• C. $(-\sqrt{2},-1)$
• D. $(-\sqrt{2},\sqrt{2})$

Answer 1

Answer: (B), (C)

$(-\sqrt{2},-1)$

Angle between pair of tangents is given by
$\tan \theta =\frac{2Lr}{L^2-r^2}=\frac{2\sqrt{S_1}}{S_1-1}=\frac{2\sqrt{a^2-1}}{a^2-2}$

If $\theta$ lies in second quadrant then $\tan \theta <0$
$\Rightarrow \frac{2\sqrt{a^2-1}}{a^2-2}<0$
$\Rightarrow a^2-1>0$ and $a^2-2<0$
$\Rightarrow |a|>1$ and $|a|<2$
$\Rightarrow a\in (-\sqrt{2},-1)\cup (1,\sqrt{2})$