Let \(P\) be a point on the ellipse \(\frac{x^2}{9}+\frac{y^2}{4}=1\). Let the line passing through \(P\) and parallel to \(y\)-axis meet the circle \(x^2+y^2=9\) at point \(Q\) such that \(P\) and \(Q\) are on the same side of the \(x\)-axis. Then, the eccentricity of the locus of the point \(R\) on \(P Q\) such that \(P R: R Q=4: 3\) as \(P\) moves on the ellipse, 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.