[Tex/LaTex] Tikz, tangent between two circles

Question

I am trying to create the image below

Here is my code

\documentclass[10pt,a4paper]{article}
\usepackage{tikz,tkz-euclide}

\begin{document}
\center
\begin{tikzpicture}
\tkzDefPoint(0,0){O}
\tkzDefPoint(0,1.25){A1}
\tkzDefPoint(0,3){A2}
\tkzDefPoint(3,0){A3}
\tkzDrawCircle[very thick,color=blue,fill=green!80!black,opacity=0.6](O,A2)
\tkzDrawCircle[fill=white](O,A1)
\end{tikzpicture}

\end{document}

My problem is, how Do I create the tangent line? And is there a more efficent way than mine? =)

Answer 1

If the tangent is horizontal, then no trig is needed and the following works.

\begin{tikzpicture}
\draw[thick,fill=green!20] (0,0) circle (2cm);
\path[clip] (0,0) circle (2cm);
\fill[green!30] (0,0) circle (2cm);
\draw[thick,blue,fill=white] (0,0) circle (1cm);
\draw[thick,red] (-2,1) -- (2,1) node[midway,above,black] {A};
\end{tikzpicture}

Note that this is pretty minimal, and just for generating the image you want. More sophisticated methods are available (see Jake's answer), but often I'll default to quick-and-dirty like this, because as much as I love making diagrams, time is money!