AHSME 1983 Problem 22

Consider the two functions $f(x) = x^2+2bx+1$ and $g(x) = 2a(x+b)$, where the variable $x$ and the constants $a$ and $b$ are real numbers. Each such pair of constants $a$ and $b$ may be considered as a point $(a,b)$ in an $ab$-plane. Let $S$ be the set of such points $(a,b)$ for which the graphs of $y = f(x)$ and $y = g(x)$ do not intersect (in the $xy$-plane). The area of $S$ is $\textbf{(A)} \ 1 \textbf{(B)} \ \pi \textbf{(C)} \ 4 \textbf{(D)} \ 4 \pi \textbf{(E)} \ \text{infinite}$