Question

ABCD is a square, in which a circle is inscribed touching all the sides of square. In the four corners of the square, 4 smaller circles of equal radii are drawn, containing the maximum possible area. What is the ratio of the area of the larger circle to that of the sum of the areas of four smaller circles?

• A. $1:(68-48\sqrt{2})$
• B. $1:17\sqrt{2}$
• C. $3:(34-12\sqrt{2})$
• D. $Noneofthese$

Answer 1

The correct option is A $1:(68-48\sqrt{2})$

Let E be the radius of the larger circle. Also, PQ = MQ = QR = r (radius of the smaller circle)
E $(\sqrt{2}-1)=r(\sqrt{2}+1)$
$\Rightarrow r=E(\sqrt{2}-1{)}^{2}r=E(3-2\sqrt{2})\therefore \frac{Areaoflargercircle}{Areaofsmallercircle}=\frac{\pi E^2}{4\pi r^2}=\frac{E^2}{4(3-2\sqrt{2}{)}^2{E}^2}=\frac{1}{4(17-12\sqrt{2})}$