एक दीर्घवृत \(E: \frac{x^2}{a^2}+\frac{y^2}{b^2}=1\) अतिपरवलय \(H: \frac{x^2}{49}-\frac{y^2}{64}=-1\) के शीर्षो से होकर जाता है। माना दीर्घवृत \(E\) के दीर्घ तथा लघु अक्ष क्रमशः अतिपरवलय \(H\) के अनुप्रस्थ तथा संयुग्मी अक्ष के सम्पाती हैं। माना \(E\) तथा \(H\) की उत्केन्द्रताओं का गुणनफल \(\frac{1}{2}\) है। यदि दीर्घवृत \(E\) की नाभिलंब जीवा की लंबाई \(l\) है, तो \(113 l\) का मान 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.