In the given figure, the ratio of the area of the largest square to that of the smallest square is:

From the figure,

$$r_2=b$$

From $$\triangle\ $$OAB,

$$r_1^2=a^2+a^2$$

$$=$$>  $$r_1^2=2a^2$$

$$=$$>  $$a^2=\frac{r_1^2}{2}$$

From $$\triangle\ $$OCD,

$$r_2^2=r_1^2+r_1^2$$

$$=$$>  $$r_2^2=2r_1^2$$

$$\therefore\ $$Ratio of the area of the largest square to that of smallest square = $$\left(2b\right)^2\ :\ \left(2a\right)^2$$ = $$b^2\ :\ a^2$$

$$=r_2^2\ :\ \frac{r_1^2}{2}$$

$$=2r_1^2\ :\ \frac{r_1^2}{2}$$

$$=2\ :\ \frac{1}{2}$$

$$=4\ :\ 1$$

Hence, the correct answer is Option A