The diameters of two circles are the side of a square and the diagonal of the square. The ratio of the areas of the smaller circle and the larger circle is

Diagonal of a square = \( \Large \sqrt{2}\times Side \)

Ratio of area of smaller circle to larger circle

\( \Large = \frac{ \pi r_{1}^{2}}{\pi r_{2}^{2}} = \frac{ \pi  \times  \left(\frac{a}{2}\right)^{2}}{ \pi  \times  \left(\frac{\sqrt{2}a}{2}\right)^{2} } \)

Here, a = Diameter of smaller circle

= \( \Large \frac{1}{2} \)=1:2