एक अतिपरवलय, जिसका अनुप्रस्थ अक्ष शांकव \(\frac{x^{2}}{3}+\frac{y^{2}}{4}=4\) के दीर्घ अक्ष की दिशा में है तथा जिसके शीर्ष इस शांकव की नाभियों पर है। यदि अतिपरवलय की उत्केन्द्रता \(\frac{3}{2}\) है, तो निम्न में से कौन सा बिंदु इस पर स्थित नहीं है ?

A \(\left( {\sqrt 5 ,2\sqrt 2 } \right)\)
B \((0, 2)\)
C \(\left( {5,2\sqrt 3 } \right)\)
D \(\left( {\sqrt 10 ,2\sqrt 3 } \right)\)

Verified Solution

Answer & Solution

Correct Answer

(C) \(\left( {5,2\sqrt 3 } \right)\)

Step-by-step Solution

Detailed explanation

\(\left( c \right)\,\,\,\,\frac{{{x^2}}}{{12}} + \frac{{{y^2}}}{{16}} = 1\) \(e = \sqrt {1 - \frac{{12}}{{16}}} = \frac{1}{2}\) Foci \((0,2)\) &\((0,-2)\) So, transverse axis of hyperbola \(a = 2b = 4 \Rightarrow b = 2\) \({a^2} = {1^2}\left( {{e^2} - 1} \right)\)…

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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