If \(\frac{x}{y}\) = \(\frac{a - \sqrt {2}}{a + \sqrt {2}}\) 

then \(\frac{x^2 + y^2}{x^2 - y^2}\) = ?

\(\frac{-a^2 + 2}{2\sqrt {2}a}\)

Let a  = multiple of \(\sqrt {2}\) for convenience →

     a = 2\(\sqrt {2}\)

so, \(\frac{x}{y}\) = \(\frac{2\sqrt {2} - \sqrt {2}}{2\sqrt {2} + \sqrt {2}}\) = \(\frac{\sqrt {2}}{3\sqrt {2}}\)

\(\frac{x}{y}\) = \(\frac{1}{3}\)

Put in and find = \(\frac{x^2 + y^2}{x^2 - y^2}\)

                      = \(\frac{1 + 9}{1 - 9}\) = \(\frac{10}{-8}\)

                      = \(\frac{-5}{4}\)

Put in option and satisfy

Take option B

\(\frac{-a^2 + 2}{2\sqrt {2}a}\)

⇒ - \(\frac{(2\sqrt {2})^2 + 2}{2\sqrt {2}× 2\sqrt {2}}\)

⇒ - \(\frac{10}{8}\) = -\(\frac{5}{4}\) satisfied