Question

The range of value of $P$ such that the angle $\theta$ between the pair of tangents drawn from the point $\left(p.0\right)$ to the circle $x^2+y^2=1$ lies in $(\frac{\pi }{3},\pi )$, is

• A. $(-2,-1)\cup (1,2)$
• B. $(-3,-2)\cup (2,3)$
• C. $(0,2)$
• D. None of these

Answer 1

Answer: (A)

$(-2,-1)\cup (1,2)$

We have $\frac{\pi }{3}<\theta <\pi$

$\Rightarrow \frac{\pi }{6}<\frac{\theta }{2}<\frac{\pi }{2}\Rightarrow \frac{1}{2}<\sin \left(\frac{\theta }{2}\right)<1$

$\Rightarrow \frac{1}{2}<\frac{1}{a}<1[\because \sin \left(\frac{\theta }{2}\right)=\frac{1}{a}]$

$\therefore 1<a<2$

There can be symmetrical points on the negative x-axis too.

Hence, we have $a\in (-2,-1)\cup (1,2)$