2006 AMC 12A Problem 21

Let

S_1=\{(x,y)|\log_{10}(1+x^2+y^2)\le 1+\log_{10}(x+y)\}

and

S_2=\{(x,y)|\log_{10}(2+x^2+y^2)\le 2+\log_{10}(x+y)\}.

What is the ratio of the area of S_2 to the area of S_1?

\mathrm{(A) \ } 98\qquad \mathrm{(B) \ } 99\qquad \mathrm{(C) \ } 100\qquad \mathrm{(D) \ } 101\qquad \mathrm{(E) \ } 102

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.