ધારો કે અતિવલય \(\frac{x^2}{16}-\frac{y^2}{9}=1\) ની ઉત્કેન્દ્રતા \(\mathrm{e}_1\) છે અને ઉપવલય \(\frac{x^2}{\mathrm{a}^2}+\frac{y^2}{\mathrm{~b}^2}=1, \mathrm{a}>\mathrm{b}\) ની ઉત્કેન્દ્રતા \(\mathrm{e}_2\) છે, ને અતિવલયની નાભીઓમાંથી પસાર થાય છે. તે \(\mathrm{e}_1 \mathrm{e}_2=1\) હોય, તો \(x\)-અક્ષને સમાંતર તથા \((0,2)\) માંથી પસાર થતી ઉપવલયની જીવાની લંબાઈ ........... છે.

A \(4 \sqrt{5}\)
B \(\frac{8 \sqrt{5}}{3}\)
C \(\frac{10 \sqrt{5}}{3}\)
D \(3 \sqrt{5}\)

Verified Solution

Answer & Solution

Correct Answer

(C) \(\frac{10 \sqrt{5}}{3}\)

Step-by-step Solution

Detailed explanation

\( H: \frac{x^2}{16}-\frac{y^2}{9}=1 \quad e_1=\frac{5}{4} \) \(\therefore e_1 e_2=1 \Rightarrow e_2=\frac{4}{5}\) Also, ellipse is passing through \(( \pm 5,0)\) \( \therefore a=5 \text { and } b=3 \) \( E: \frac{x^2}{25}+\frac{y^2}{9}=1\) End point of chord are…

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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