6. The function in the form of $y=\frac{b}{|x|-a}(a, b>0)$

Resembles the Chinese character “唄”, it is called the “唄 function”, and the point symmetric to the intersection with the $y$-axis about the origin is called the “目 point”. A circle with the “目 point” as its center and that has common points with the “唄 function” is called a “圆圆”. When $a=b=1$, the minimum area of all “圆圆” is . $\qquad$
6. $3 \pi$.

As shown in Figure 3, when $a=b=1$, the analytical expression of the function is $y=\frac{1}{|x|-1}$, and its intersection point with the $y$-axis is $(0,-1)$. Thus, the point $C(0,1)$, and the distance between them is 2.

Take any point $(x, y)$ on the graph of the function in the first quadrant, the distance to the fixed point is
$$
\begin{array}{l}
d=\sqrt{x^{2}+(y-1)^{2}}=\sqrt{x^{2}+\left(\frac{1}{x-1}-1\right)^{2}} \\
=\sqrt{\left(x-\frac{1}{x-1}\right)^{2}+3} \geqslant \sqrt{3} .
\end{array}
$$

The equality holds if and only if $x=\frac{1+\sqrt{5}}{2}$. Therefore, the minimum value of the area in the circle is $3 \pi$.