माना कि \(e_1\) और \(e_2\) दीर्घवृत्त \(\frac{\mathrm{x}^2}{\mathrm{~b}^2}+\frac{\mathrm{y}^2}{25}=1 \quad\) और अतिपरवलय \(\quad \frac{\mathrm{x}^2}{16}-\frac{\mathrm{y}^2}{\mathrm{~b}^2}=1\), क्रमशः की उत्केंद्रताएं हैं। यदि \(\mathrm{b} \lt 5\) और \(\mathrm{e}_1 \mathrm{e}_2=1\), तो उस दीर्घवृत्त की उत्केंद्रता क्या होगी जिसके अक्ष निर्देशांक अक्षों के अनुदिश हैं और जो सभी चार नाभियों (दीर्घवृत्त की दो और अतिपरवलय की दो) से होकर गुजरता है?

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.