Question

Suppose we have two circles of radius $2$ each in the plane such that the distance between their centres is $2\sqrt{3}$. The area of the region common to both circles lies between.

• A. $0.5$ and $0.6$
• B. $0.65$ and $0.7$
• C. $0.7$ and $0.75$
• D. $0.8$ and $0.9$

Answer 1

Answer: (C)

$0.7$ and $0.75$

Let point of intersection be A and B and centers be O and N.

Let AB intersect ON at M. AB is perpendicular to ON and bisects ON.

$OM=\sqrt{3}$

$OA=2$

$cos\left(\angle AOM\right)=\frac{\sqrt{3}}{2}$

$\angle AOM=30degree$

$\angle AOB=60degree$

Area of common region $=2\times [areaofsectorAOB-areaof△AOB]$

$=2\times \left[\right(\frac{1}{6}\times \pi \times 2^2)-\frac{\sqrt{3}}{4}\times 2^2]$

$=8\times [\frac{\pi }{6}-\frac{\sqrt{3}}{4}]$

$=8\times 0.0905=0.72$

$\therefore$ Answer is C.