माना रेखा \(\mathrm{L}: \sqrt{2} \mathrm{x}+\mathrm{y}=\alpha\), वृत्त \(\mathrm{x}^2+\mathrm{y}^2=3\) तथा परवलय \(x^2=2 y\) के प्रतिच्छेदन बिन्दु \(\mathrm{P}\) (प्रथम चतुर्थांश में) से होकर जाती है। माना रेखा \(\mathrm{L}\) बराबर त्रिज्या \(2 \sqrt{3}\) के दो वृत्तों \(\mathrm{C}_1\) तथा \(\mathrm{C}_2\) को स्पर्श करती है। यदि वृत्तों \(\mathrm{C}_1\) तथा \(\mathrm{C}_2\) के केन्द्र \(\mathrm{Q}_1\) तथा \(\mathrm{Q}_2, \mathrm{y}\)-अक्ष पर हैं तो त्रिभुज \(\mathrm{PQ}_1 \mathrm{Q}_2\) के क्षेत्रफल का वर्ग ........... 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.