\(\sqrt{2000}\)=\(xy+\sqrt{\left(1+x^2\right)\left(1+y^2\right)}\)
\(\Rightarrow2000=x^2y^2+\left(1+x^2\right)\left(1+y^2\right)+2xy\sqrt{\left(1+y^2\right)\left(1+x^2\right)}\)
=\(x^2y^2+1+x^2+y^2+x^2y^2+2xy\sqrt{\left(1+x^2\right)\left(1+y^2\right)}\)
\(\Rightarrow x^2+y^2+2x^2y^2+2xy\sqrt{\left(1+x^2\right)\left(1+y^2\right)}=2000-1=1999\)
ma \(S^2=x^2\left(1+y^2\right)+y^2\left(1+y^2\right)+2xy\sqrt{\left(1+x^2\right)\left(1+y^2\right)}\)
=\(x^2+x^2y^2+y^2+x^2y^2+2xy\sqrt{\left(1+x^2\right)\left(1+y^2\right)}\)
=\(x^2+y^2+2x^2y^2+2xy\sqrt{\left(1+x^2\right)\left(1+y^2\right)}\) =\(1999\Rightarrow S=\sqrt{1999}\)