मान \(a\) और \(b , \frac{1}{ a ^{2}}+\frac{1}{ b ^{2}}=\frac{1}{4}\) को संतुष्ट करने वाली दो संख्याएँ हैं, तो चररेखा, \(\frac{x}{ a }+\frac{y}{ b }=1\) मूल बिंदु से डाले गए लंब का पाद, स्थित है

A एक अतिपरवलय पर, जिसका प्रत्येक अर्ध अक्ष \(=\sqrt{2}\) है।
B एक अतिरपरवलय पर, जिसका प्रत्येक अर्ध अक्ष \(=2\) है।
C एक वृत्त पर, जिसकी त्रिज्या \(=2\) है।
D एक वृत्त पर, जिसकी त्रिज्या \(=\sqrt{2}\) है।

Verified Solution

Answer & Solution

Correct Answer

(C) एक वृत्त पर, जिसकी त्रिज्या \(=2\) है।

Step-by-step Solution

Detailed explanation

Let the foot of the perpendicular from \((0,0)\) on the variable line \(\frac{x}{a} + \frac{y}{b} = 1\,\) is \(\left( {{x_1} > {y_1}} \right)\) Hence, perpendicular distance of the variable line \(\frac{x}{a} + \frac{y}{b} = 1\,\,\) from the point \(O\)…

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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