माना \(S =\left\{( x , y ) \in R ^{2}: \frac{ y ^{2}}{1+ r }-\frac{ x ^{2}}{1- r }=1\right\}\), जहाँ \(r \neq \pm 1\) है। तब \(S\)

A एक अतिपरवलय को दर्शाता है जिसकी उत्केन्द्रता \(\frac{2}{\sqrt{1- r }}\), जब \(0 < r < 1\) है।
B एक दीर्घवृत्त को दर्शाता है जिसकी उत्केन्द्रता \(\sqrt{\frac{2}{ r +1}}\), जब \(r > 1\) है।
C एक अतिपरवलय को दर्शाता है जिसकी उत्केन्द्रता \(\frac{2}{\sqrt{ r +1}}\), जहाँ \(0 < r < 1\) है।
D एक दीर्घवृत्त को दर्शाता है जिसकी उत्केन्द्रता \(\frac{1}{\sqrt{ r +1}}\), जहाँ \(r > 1\) है।

Verified Solution

Answer & Solution

Correct Answer

(B) एक दीर्घवृत्त को दर्शाता है जिसकी उत्केन्द्रता \(\sqrt{\frac{2}{ r +1}}\), जब \(r > 1\) है।

Step-by-step Solution

Detailed explanation

\(\frac{{{y^2}}}{{1 + r}} - \frac{{{x^2}}}{{1 - r}} = 1\) \(r > 1\,\,\,\, \Rightarrow \) ellipse \(e = \sqrt {1 - \left( {\frac{{r - 1}}{{r + 1}}} \right)} = \sqrt {\frac{2}{{r + 1}}} \)

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.

Same subject

Explore more questions on app

From JEE Mains

Explore more questions on app