Five charges, '\(q\)' each are placed at the comers of a regular pentagon of side '\(a\)' as shown in figure. First, charge from '\(A\)' is removed with other charges intact, then charge at '\(A\)' is replaced with an equal opposite charge. The ratio of magnitudes of electric fields at \(\mathrm{O}\), without charge at \(A\) and that with equal and opposite charge at \(A\) 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.