Question

Let $S_1$ be the sum of areas of the squares whose sides are parallel to coordinate axes.
Let $S_2$ be the sum of areas of the slanted squares as shown in the figure. Then $S_1/S_2$ is

• A. $2$
• B. $\sqrt{2}$
• C. $1$
• D. $\frac{1}{\sqrt{2}}$

Answer 1

The correct option is A $2$


$S_1=a^2+{\left(\frac{a}{2}\right)}^2+{\left(\frac{a}{4}\right)}^2+........\infty =\frac{a^2}{1-\frac{1}{4}}=\frac{4a^2}{3}$

$S_1={\left(\frac{a}{\sqrt{2}}\right)}^2+{\left(\frac{a}{2\sqrt{2}}\right)}^2+{\left(\frac{a}{4\sqrt{2}}\right)}^2+........\infty =\frac{{\left(\frac{a}{\sqrt{2}}\right)}^2}{1-\frac{1}{4}}=\frac{2a^2}{3}$

Hence $S_1/S_2=2$