माना एक दीर्घवृत्त, जिसका दीर्घ-अक्ष \(X\)-अक्ष के अनुदिश है तथा केंद्र मूलबिन्दु पर है, के नाभिलम्ब की लंबाई \(8\) है। यदि दीर्घवृत्त की नाभियों के बीच की दूरी, इसके लघु-अक्ष की लंबाई के समान हो, तो निम्न में से कौन-सा बिन्दु इस पर स्थित है ?

A \(\left( {4,\sqrt 2 ,2\sqrt 2 } \right)\)
B \(\left( {4,\sqrt 3 ,2\sqrt 2 } \right)\)
C \(\left( {4,\sqrt 3 ,2\sqrt 3 } \right)\)
D \(\left( {4,\sqrt 2 ,2\sqrt 3 } \right)\)

Verified Solution

Answer & Solution

Correct Answer

(B) \(\left( {4,\sqrt 3 ,2\sqrt 2 } \right)\)

Step-by-step Solution

Detailed explanation

Consider \(\frac{{{x^2}}}{{{a^2}}} + \frac{{{y^2}}}{{{b^2}}} = 1\) Given that \(2b = 2ae\) \( \Rightarrow b = ae\) and \(\frac{{2{b^2}}}{a} = 8\) \(a\left( {1 - {e^2}} \right) = 4,{a^2}{e^2} = {a^2}\left( {1 - {e^2}} \right)\) \( \Rightarrow {e^2} = \frac{1}{2}\)…

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.

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