Let \(S\) and \(S^{\prime}\) be the foci of an ellipse \(E\) and \(B\) be one end of its minor axis. Let \(\left\lfloor\mathrm{S}^{\prime} \mathrm{SB}=\frac{\pi}{6}\right.\) and \((2 \sqrt{3}, 1)\) be a point on \(\mathrm{E}\). If \(\mathrm{X}\)-axis is the major axis and \(\mathrm{Y}\)-axis is the minor axis of the ellipse \(E\), then the sum of the squares of the lengths of major axis and minor axis 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.