12. Given the functions $f(x)=x^{2}+2 b x+1$ and $g(x)=2 a(x+b)$, where $x, a, b$ are real numbers, such that the graphs of $y=$ $f(x)$ and $y=g(x)$ do not intersect in the $x o y$ plane, the set of real number pairs $(a, b)$ that satisfy this condition is denoted as $A$. Then, the area of the figure $T$ represented by $A$ in the $a o b$ plane is $\qquad$
12. $\pi$ factor pairs $(a, b)$ make the equation $x^{2}+2 b x+1=2 a(x+b)$ have no real roots, $\Delta<0, a^{2}+b^{2}<1$, which represents the interior of a circle.