Question

There is a square of side $4$ unit. A point in the interior of the square is randomly chosen and a circle of radius $1$ unit is drawn centered at the point. Then probability that the circle intersects the square exactly twice is

• A. $\frac{\pi +4}{16}$
• B. $\frac{\pi +6}{16}$
• C. $\frac{\pi +8}{16}$
• D. $\frac{\pi }{16}$

Answer 1

The correct option is C $\frac{\pi +8}{16}$

Case $\text{I}:$ When the circle intersects the square exactly twice where the two points are on different sides.


Favourable area $=4\times \left(\frac{1}{4}\pi \right(1{)}^2)$

Case $\text{II}:$ When the circle intersects the square exactly twice where the two points are on the same side.


Favourable area $=(1\times 2)\times 4=8$
$\therefore$ Required probability $=\frac{\pi +8}{16}$