2024 AMC 10B Problem 21

Two straight pipes (circular cylinders), with radii 1 and \tfrac14, lie parallel and in contact on a flat floor. The figure below shows a head-on view. What is the sum of the possible radii of a third parallel pipe lying on the same floor and in contact with both?

\textbf{(A)} \dfrac{1}{9} \qquad \textbf{(B)} 1 \qquad \textbf{(C)} \dfrac{10}{9} \qquad \textbf{(D)} \dfrac{11}{9} \qquad \textbf{(E)} \dfrac{19}{9}

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.