The area of the region included between the curves x ^ 2 + y ^ 2 =...

Given curves are

$\sqrt{\left|x\right|} + \sqrt{\left|y\right|} = \sqrt{a} . . . \left(i\right)$

and ${x}^{2} + {y}^{2} = {a}^{2} . . . \left(ii\right)$

Now, required areas $= 4$[shaded area in the first quadrant]

$= 4\left[\dfrac{\pi{a}^{2}}{4} - {∫}_{0}^{a}{\left(\sqrt{a} - \sqrt{x}\right)}^{2}dx\right]$ [from Eqs. (i) and (ii)]

$= 4\left[\dfrac{\pi{a}^{2}}{4} - {∫}_{0}^{a}\left(a + x - 2\sqrt{a}\sqrt{x}\right)dx\right]$

$= 4\dfrac{\pi{a}^{2}}{4} - 4{\left[ax + \dfrac{{x}^{2}}{2} - \dfrac{4}{3}\sqrt{a}{x}^{3 / 2}\right]}_{0}^{a}$

$= \left(\pi - \dfrac{2}{3}\right){a}^{2}$ sq units

Hence, option (a) is correct.