If $$xy=\frac{a+2}{3}$$ and $$\frac{x}{y}=\frac{1}{3}$$, then find the value of $$\frac{x^{2}+y^{2}}{x^{2-}y^{2}}$$

Given : $$xy=\frac{a+2}{3}$$ and $$\frac{x}{y}=\frac{1}{3}$$

Multiplying both equations, we get : $$x^2=\frac{a+2}{9}$$ -------------(i)

Dividing both equations, => $$y^2=a+2$$ -------------(ii)

Adding equations (i) and (ii),

=> $$x^2+y^2=(\frac{a+2}{9})+(a+2)=\frac{9a+18+a+2}{9}$$

=> $$x^2+y^2=\frac{10a+20}{9}$$ -----------(iii)

Similarly, subtracting equation (ii) from (i), => $$x^2-y^2=\frac{-8a-16}{9}$$ -------------(iv)

Dividing equation (iii) by (iv), we get :

=> $$\frac{x^2+y^2}{x^2-y^2}=(\frac{10a+20}{9})\div(\frac{-8a-16}{9})$$

= $$\frac{10a+20}{-8a-16}=\frac{10(a+2)}{-8(a+2)}$$

= $$\frac{-5}{4}$$

=> Ans - (D)