Description for Correct answer:
\( \Large x = \frac{\sqrt{2}+1}{\sqrt{2}-1} \)

=> \( \Large x = \frac{\sqrt{2}+1}{\sqrt{2}-1} \times \frac{\sqrt{2}+1}{\sqrt{2}-1} \)

\( \Large \frac{2 + 1 + 2\sqrt{2}}{1} = 3 + 2\sqrt{2} \)

\( \Large x = 3 + 2\sqrt{2} \) ...(i)

and \( \Large x - y = 4\sqrt{2} \)

=> \( \Large y = x - 4\sqrt{2} = 3 + 2\sqrt{2} - 4\sqrt{2} \)

[Frm Eq. (i)]

\( \Large = 3 - 2\sqrt{2} \)

Now, \( \Large x^{2} + y^{2} = \left(3+2\sqrt{2}\right)^{2} + \left(3 - 2\sqrt{2}\right)^{2} \)

\( \Large = 9 + 8 + 12\sqrt{2} + 9 + 8 - 12\sqrt{2} \)

= 34