question_answer

In the adjoining diagram ABCD is a square with side 'a' cm. In the diagram the area of the larger circle with centre 'O' is equal to the sum of the areas of all the rest four circles with equal radii, whose centers are P, Q, R and S. What is the ratio between the diagonal of square and radius of a smaller circle?

A)  \[\left( 2\sqrt{2}+3 \right)\]                    

B)  \[\left( 2+3\sqrt{2} \right)\]

C)  \[\left( 4+3\sqrt{2} \right)\]                    

D)  can't be determined

Correct Answer: B

Solution :

(b): Let the radius of each smaller circle be 'r' and radius of the larger circle be ?R?, then \[OR=OP=R+r=3r\];     \[\pi {{R}^{2}}=4\pi {{r}^{2}}\Rightarrow R=2r\] Also, PM = r   (PM is the perpendicular on AB) \[AP=\sqrt{2}r\];  \[AO=AP+PO=r\sqrt{2}+3r=r\left( 3+\sqrt{2} \right)\] \[AC=2AO=2r\left( 3+\sqrt{2} \right)\], which is the diagonal \[\therefore \] Required ratio \[=\frac{2r\left( 3+\sqrt{2} \right)}{\sqrt{2}r}=\left( 2+3\sqrt{2} \right)\]