ધારો કે વક્રો \(4\left(x^{2}+y^{2}\right)=9\) અને \(y^{2}=4 x\) ના સામાન્ય સ્પર્શકો \(Q\) બિંદુમાં છેદે છે. ધારે કે \(O\) કેન્દ્રવાળા એક ઉપવલયના ગૌણ અક્ષ અને પ્રધાન અક્ષ ની અર્લંધબાઈઓ અનુક્રમે \(OQ\) અને \(6\) છે.જો આ ઉપવલય ઉત્કેન્દ્રતા \(e\) અને નાભિલંબની લંબાઈ \(l\) હોય, તો \(\frac{l}{ e ^{2}}=\dots\dots\dots\)

A \(5\)
B \(4\)
C \(3\)
D \(2\)

Verified Solution

Answer & Solution

Correct Answer

(B) \(4\)

Step-by-step Solution

Detailed explanation

\(x^{2}+y^{2}=\frac{9}{4} \quad y=4 x\) Equation tangent in slope form \(y=m x \pm \frac{3}{2} \sqrt{\left(1+m^{2}\right)}\) .....\((1)\) \(y=m x+\frac{1}{m}\) .....\((2)\) compare \((1)\) and \((2)\) \(\pm \frac{3}{2} \sqrt{\left(1+m^{2}\right)}=\frac{1}{m^{2}}\)…

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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વર્ગ	\(0-4\)	\(4-8\)	\(8-12\)	\(12-16\)	\(16-20\)
આવ્રુતિ	\(3\)	\(9\)	\(10\)	\(8\)	\(6\)