How to fill area between two ellipses and two lines?

Question

Actually a question. I can not find a simple and elegant answer. The use of arcs makes me despondent.

\documentclass[tikz, border=1cm]{standalone}
\usetikzlibrary{intersections}

\begin{document}
    \begin{tikzpicture}
        \draw[name path=circles] (0,0) circle (4 and 2) circle (3 and 1);
        \path[name path=line1] (0,0) -- (10:5);
        \path[name path=line2] (0,0) -- (-10:5);
        \draw[name intersections={of=circles and line1}] (0,0) -- (intersection-1);
        \draw[name intersections={of=circles and line2}] (0,0) -- (intersection-1);
    \end{tikzpicture}
\end{document}

Answer 1

\documentclass[tikz, border=1cm]{standalone}
\begin{document}
\begin{tikzpicture}
\begin{scope}
\clip  (0,0) -- (10:5) -- (-10:5) -- cycle;
\fill[pink, even odd rule] (0,0) ellipse[x radius=4, y radius=2] ellipse[x radius=3, y radius=1];
\end{scope}
\draw (0,0) ellipse[x radius=4, y radius=2] ellipse[x radius=3, y radius=1];
\draw (0,0) -- (10:5) (0,0) -- (-10:5);
\end{tikzpicture}
\end{document}

Answer 2

Here's an answer using the spath3 library, which works by splitting the elliptical and line paths where they intersect and then constructing a new path by welding together pieces of the resulting paths.

\documentclass{article}
%\url{https://tex.stackexchange.com/q/652890/86}
\usepackage{tikz}
\usetikzlibrary{intersections,spath3}

\begin{document}
\begin{tikzpicture}
\path[spath/save=circles] (0,0)
circle[x radius=4, y radius=2]
circle[x radius=3, y radius=1]
;
% the overlay key means that these don't add to the picture size,
% also, a single path here simplifies the code later
\path[spath/save=lines,overlay] (10:5) -- (0,0) -- (-10:5);

% split the paths where they overlap each other
\tikzset{
  spath/split at intersections={circles}{lines},
  % save the components of the paths
  spath/get components of={circles}\circlesCpts,
  spath/get components of={lines}\linesCpts,
}

\fill[
  magenta,
  % takes a little experimenting to get the order right ...
  spath/use={\getComponentOf\circlesCpts{3}},
  spath/use={\getComponentOf\linesCpts{2},weld},
  spath/use={\getComponentOf\circlesCpts{6},reverse,weld},
  spath/use={\getComponentOf\linesCpts{4},weld},
];

% draw the ellipses now so that they are over the top of the
% filled region
\draw[ultra thick, spath/use={circles}];

\end{tikzpicture}
\end{document}