रेखाओं \(t x-2 y-3 t =0\); \(x-2 t y+3=0( t \in R )\) के प्रतिच्छेदन बिंदु का बिंदु पथ है

A एक दीर्घवृत्त जिसकी उत्केन्द्रता \(\frac{2}{\sqrt{5}}\) है
B एक दीर्घवृत्त जिसके दीर्घ अक्ष की लंबाई \(6\) है
C एक अतिपरवलय जिसकी उत्केन्द्रता \(\sqrt{5}\) है
D एक अतिपरवलय जिसके संयुग्मी अक्ष (conjugate axis) की लंबाई \(3\) है

Verified Solution

Answer & Solution

Correct Answer

(D) एक अतिपरवलय जिसके संयुग्मी अक्ष (conjugate axis) की लंबाई \(3\) है

Step-by-step Solution

Detailed explanation

Here, \(tx - 2y - 3t = 0\,\ and \,x - 2ty + 3 = 0\) On solving we get; \(y = \frac{{3t}}{{{t^2} - 1}}\,\,\ and \,\,x = \frac{{3{t^2} + 3}}{{{t^2} - 1}}\) Put \(t = \tan \,\theta \) \(\therefore x = - 3\,\sec 2\theta \,\ and \,\,2y = 3\left( { - tan2\theta } \right)\)…

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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