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  • Let <img alt="\mathrm{H} \mathrm{H}_{\mathrm{n}}: \frac{\mathrm{x}^2}{1+\mathrm{n}}-\frac{\mathrm{y}^2}{3+\mathrm{n}}=1, \mathrm{n} \in \mathrm{N} ." src="https://entrancecorner.oncodecogs.com

Let \mathrm{H} \mathrm{H}_{\mathrm{n}}: \frac{\mathrm{x}^2}{1+\mathrm{n}}-\frac{\mathrm{y}^2}{3+\mathrm{n}}=1, \mathrm{n} \in \mathrm{N} . Let k be the smallest even value of n such that the eccentricity of 

of H_k is a rational number. If l is the length of the latus rectum of H_k, then 21 l is equal to ____:

Option: 1

306


Option: 2

-


Option: 3

-


Option: 4

-


Answers (1)

best_answer

\begin{aligned} & \mathrm{Hn} \Rightarrow \frac{\mathrm{x}^2}{1+\mathrm{n}}-\frac{\mathrm{y}^2}{3+\mathrm{n}}=1 \\ & \mathrm{e}=\sqrt{1+\frac{\mathrm{b}^2}{\mathrm{a}^2}}=\sqrt{1+\frac{3+\mathrm{n}}{1+\mathrm{n}}}=\sqrt{\frac{2 \mathrm{n}+4}{\mathrm{n}+1}} \\ & \mathrm{e}=\sqrt{\frac{2 \mathrm{n}+4}{\mathrm{n}+1}} \\ & \mathrm{n}=48(\text { smallest even value for which } \mathrm{e} \in \mathrm{Q}) \\ & \mathrm{e}=\frac{10}{7} \\ & \mathrm{a}^2=\mathrm{n}+1 \quad \quad \quad \mathrm{~b}^2=\mathrm{n}+3 \\ & =49 \quad=51 \\ & 1=\text { length of } \mathrm{LR}=\frac{2 \mathrm{~b}^2}{\mathrm{a}} \\ & \mathrm{L}=2 \cdot \frac{51}{7} \\ & 1=\frac{102}{7} \\ & 21 \ell=306 \end{aligned}

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