How can I draw a tangent ending smoothly in a circle?

Question

I'm trying to draw a tanget on a circle, but I'm not happy with the result I get from the tangent option.

particularly, I'd like to have the lines ending smoothly in the circle. The problem is shown enlarged here:

Here is the code for this mininmal example

\documentclass{standalone}
\usepackage{tikz}
\usetikzlibrary{calc}

\begin{document}
  \begin{tikzpicture}[thick]
    \node[draw,circle,xshift=2.2cm] (big) [minimum size=25mm] {};
    \node[draw,circle] (small) [minimum size=2mm] {};
    \draw (small.south) -- (tangent cs:node=big,point={(small.south)});
    \draw (small.north) -- (tangent cs:node=big,point={(small.north)},solution=2);
   \end{tikzpicture}
\end{document}

I'd greatly appreciate any advice allowing me to do this!

Answer 1

You already have the solution: Just apply the same to the small circle, and throw some outer sep=0 for a nice blend.

\documentclass{standalone}
\usepackage{tikz}
\usetikzlibrary{calc}

\begin{document}

  \begin{tikzpicture}[thick]
    \node[draw,circle,xshift=2.2cm,minimum size=25mm,outer sep=0] (big) {};
    \node[draw,circle,minimum size=2mm,outer sep=0] (small) {};
    \draw (tangent cs:node=small,point={(big.south)},solution=2) -- (tangent cs:node=big,point={(small.south)});
    \draw (tangent cs:node=small,point={(big.north)},solution=1) -- (tangent cs:node=big,point={(small.north)},solution=2);
   \end{tikzpicture}

\end{document}

Answer 2

Here is an exact solution (via barycentric coordinate system as in this answer to the question Can PSTricks or others draw the 4 common tangent lines of 2 “disjoint” circles without having to do extra calculations?):

\documentclass{standalone}
\usepackage{tikz}
\usetikzlibrary{calc}
\begin{document}
\begin{tikzpicture}[inner sep=0,outer sep=0]
  % radii
  \pgfmathsetmacro{\rbig}{20mm}
  \pgfmathsetmacro{\rsmall}{1mm}
  % the two circles
  \node[draw,circle,xshift=\rsmall+\rbig+1mm,minimum size=2*\rbig pt] (big) {};
  \node[draw,circle,minimum size=2*\rsmall pt] (small) {};
  % the good point !
  \coordinate (c) at (barycentric cs:big=-\rsmall,small=\rbig);
  \fill[red](c) circle (.2pt);
  % the two tangents
  \draw (tangent cs:node=small,point={(c)},solution=2) -- (tangent cs:node=big,point={(c)},solution=2);
  \draw (tangent cs:node=small,point={(c)},solution=1) -- (tangent cs:node=big,point={(c)},solution=1);
\end{tikzpicture}
\end{document}

Edit:

The following code computes the difference between Percusse's solution (a good approximation) and this solution (an "exact" solution):

\documentclass{standalone}
\usepackage{tikz}
\usetikzlibrary{calc}
\begin{document}
\begin{tikzpicture}[inner sep=0,outer sep=0]
  \pgfmathsetmacro{\rbig}{20mm}
  \pgfmathsetmacro{\rsmall}{1mm}
  \node[draw,circle,xshift=\rsmall+\rbig+1mm,minimum size=2*\rbig pt] (big) {};
  \node[draw,circle,minimum size=2*\rsmall pt] (small) {};
  \coordinate (c) at (barycentric cs:big=-\rsmall,small=\rbig);

  \coordinate (exact small 1) at (tangent cs:node=small,point={(c)},solution=1);
  \coordinate (approx small 1) at (tangent cs:node=small,point={(big.south)},solution=2);
  \coordinate (exact big 1) at (tangent cs:node=big,point={(c)},solution=1);
  \coordinate (approx big 1) at (tangent cs:node=big,point={(small.south)},solution=1);

  % the difference
  \path let \p1=($(exact small 1) - (approx small 1)$),
  \p2=($(exact big 1) - (approx big 1)$) in
  \pgfextra{
    \typeout{small circle difference:\x1,\y1}
    \typeout{big circle difference:\x2,\y2}
  };
\end{tikzpicture}
\end{document}

And from the log:

small circle difference:-0.6035pt,0.73969pt
big circle difference:1.43744pt,-2.58029pt