Question

The range of values of $\lambda (\lambda >0)$ such that the angle $\theta$ between the pair of tangents drawn from $(\lambda ,0)$ to the circle $x^2+y^2=4$ lies in $(\frac{\pi }{2},\frac{2\pi }{3})$ is

• A. $(\frac{4}{\sqrt{3}},2\sqrt{2})$
• B. $(0,\sqrt{2})$
• C. $(1,2)$
• D. None of these

Answer 1

Answer: (A)

$(\frac{4}{\sqrt{3}},2\sqrt{2})$

We have $\frac{\pi }{2}<\theta <\frac{2\pi }{3},$ i.e. $\frac{\pi }{4}<\frac{\theta }{2}<\frac{\pi }{3}$
$\Rightarrow \frac{1}{\sqrt{2}}<\sin \frac{\theta }{2}<\frac{\sqrt{3}}{2}$
But, $\sin \frac{\theta }{2}=\frac{2}{\lambda }\Rightarrow \frac{1}{\sqrt{2}}<\frac{2}{\lambda }<\frac{\sqrt{3}}{2}$
$\Rightarrow \frac{4}{\sqrt{3}}<\lambda <2\sqrt{2}$