Question 22
Consider the two functions $$
\begin{aligned} f(x)&=x^2+2bx+1,\\ g(x)&=2a(x+b), \end{aligned}
$$ 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
\begin{aligned} f(x)&=x^2+2bx+1,\\ g(x)&=2a(x+b), \end{aligned}
$$ 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