Three circular discs of same material and same thickness of radii \(r, 2 r\) and \(3 r\) are placed on a horizontal plane such that their centres lie along a straight line. The radius of the middle disc is \(2 \mathrm{r}\) and it touches the other two discs. The distance of the centre of mass of the system from the centre of the smaller disc is ________ .

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.