question_answer

There are two circles intersecting each other. Another smaller circle with centre O is lying between the common regions of two larger circles. Centres of the circle (i.e., A, O and B) are lying on a straight line. If AB = 16 cm and the radii of the larger circles are 10 cm each, what is the area of the smaller circle?

A)  \[4\pi \text{ }c{{m}^{2}}\]   

B)  \[2\pi \text{ }c{{m}^{2}}\]        

C)  \[\frac{4}{\pi }\text{ }c{{m}^{2}}\]                

D)  \[\frac{\pi }{4}\text{ }c{{m}^{2}}\]

Correct Answer: A

Solution :

(a): \[AB=16cm\] \[AQ=10\text{ }cm\] and \[AO=\frac{AB}{2}=8\,cm\] \[OQ=AQ-AO\Rightarrow 10-8=2\,cm\] OQ (= OP) is the radius of the smaller enclosed circle between two arcs. Area of circle with centre O is \[\pi \times {{(2)}^{2}}=4\pi \,c{{m}^{2}}\]