Question

In the figure assume that the circle are mutually tangent and that the circle with centre at A has radius 1. The lines AB and AC are tangent to the circle with centre O at B and C respectively. If the area of the shaded region is $\pi /4$, the radius of the circle centered at O is

• A. $r=\sqrt{2}-1$
• B. $r=2-\sqrt{2}$
• C. $r=1+\sqrt{2}$
• D. $r=2+\sqrt{2}$

Answer 1

The correct option is C $r=1+\sqrt{2}$

Let, $\angle CAB=\theta$
Then, area of sector, $\frac{\theta }{360}\pi (1{)}^2=\frac{\pi }{4}$
$\theta ={90}^{\circ }$
Now, the line joining the centre of the circle bisects the angle formed between two tangets, hence, $\angle CAO={45}^{\circ }$
Now, In $△AOC$
$\angle ACO={90}^{\circ }$ (Angle between the tangent and the radius)
$\sin 45=\frac{OC}{AO}$
$\frac{1}{\sqrt{2}}=\frac{r}{r+1}$
$r+1=r\sqrt{2}$
$r=\frac{1}{\sqrt{2}-1}$ (Rationalizing)
$r=\sqrt{2}+1$