Playing around a little with scopes, line thickness and putting nodes to the ends of tangents:
\documentclass[tikz,border=2mm]{standalone}
\usetikzlibrary{calc, arrows}
\begin{document}
\begin{tikzpicture}
\node[draw,circle,xshift=2.2cm,minimum size=25mm,outer sep=0] (bigg) {};
\node[draw,circle,minimum size=2mm,outer sep=0] (smalll) {};
\draw[thick] (tangent cs:node=smalll,point={(bigg.south)},solution=2) node (a) {} -- (tangent cs:node=bigg,point={(smalll.south)}) node (b) {};
\draw[thick] (tangent cs:node=smalll,point={(bigg.north)},solution=1) node (c) {} -- (tangent cs:node=bigg,point={(smalll.north)},solution=2) node (d) {};
\begin{scope}
\clip ($(d)+(0,0.25)$) rectangle ($(b)+(2.1,-0.25)$);
\node[draw,circle,minimum size=25mm,outer sep=0,thick] at (bigg) {};
\end{scope}
\begin{scope}
\clip ($(c)+(0,0.1)$) rectangle ($(a)+(-0.2,-0.1)$);
\node[draw,circle,minimum size=2mm,outer sep=0,thick] at (smalll) {};
\end{scope}
\draw[densely dashed, red, -latex] ($(c)!0.3!(d)$) -- ($(c)!0.7!(d)$);
\draw[ultra thick, blue, -stealth] ($(b)!0.45!(a)$) -- ($(b)!0.75!(a)$);
\end{tikzpicture}
\end{document}
Output
