jmc

We have that $$\begin{array}{rl}\frac{1}{a_{n+1}}+\frac{1}{b_{n+1}}& =\frac{1}{a_n+b_n+\sqrt{a_n^2+b_n^2}}+\frac{1}{a_n+b_n-\sqrt{a_n^2+b_n^2}}\\ & =\frac{a_n+b_n-\sqrt{a_n^2+b_n^2}+a_n+b_n+\sqrt{a_n^2+b_n^2}}{(a_n+b_n{)}^2-(a_n^2+b_n^2)}\\ & =\frac{2a_n+2b_n}{2a_n{b}_n}\\ & =\frac{1}{a_n}+\frac{1}{b_n}.\end{array}$$Thus, $\frac{1}{a_n}+\frac{1}{b_n}$ is a constant, which means $$\frac{1}{a_{2012}}+\frac{1}{b_{2012}}=\frac{1}{a_0}+\frac{1}{b_0}=\boxed{\frac{1}{2}}.$$