Let H _{ n }=frac{ x ^2}{1+ n }-frac{ y ^2}{3+ n }=1, n ∈ N . Let k be smallest even value of n su

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10-2. Parabola, Ellipse, Hyperbola

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Let $H _{ n }=\frac{ x ^2}{1+ n }-\frac{ y ^2}{3+ n }=1, n \in N$. Let $k$ be the smallest even value of $n$ such that the eccentricity of $H _{ k }$ is a rational number. If $l$ is length of the latus return of $H _{ k }$, then $21 l$ is equal to $.......$.

A

$305$

B

$306$

C

$304$

D

$303$

(JEE MAIN-2023)

Solution

$Hn \Rightarrow \frac{ x ^2}{1+ n }-\frac{ y ^2}{3+ n }=1$

$e =\sqrt{1+\frac{ b ^2}{ a ^2}}=\sqrt{1+\frac{3+ n }{1+ n }}=\sqrt{\frac{2 n +4}{ n +1}}$

$e =\sqrt{\frac{2 n +4}{ n +1}}$

$n =48(\text { smallest even value for which } e \in Q )$

$e =\frac{10}{7}$

$a ^2 =n+1 \quad b ^2=n+3$

$=49 \quad, \quad=51$

$1 =\text { length of } LR =\frac{2 b ^2}{ a }$

$L =2 \cdot \frac{51}{7}$

$1 =\frac{102}{7}$

$21 \ell=306$

Standard 11

Mathematics

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