Let H n: (x²)/(1+n)-(y²)/(3+n)=1, n N. Let k be the smallest even value of n such that the eccentricity 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 _ _ _ _ _ _.

Coordinate Geometry: JEE Main 2023 Mathematics question

JEE Main 2023 · 11 Apr, Morning shift · Q35

JEE Main 202311 Apr, Morning shiftMathematicsMediumCoordinate Geometry478× since 2002

Let

\mathrm{H}_{\mathrm{n}}: \frac{x^{2}}{1+n}-\frac{y^{2}}{3+n}=1, n \in N

. Let

\mathrm{k}

be the smallest even value of

\mathrm{n}

such that the eccentricity of

\mathrm{H}_{\mathrm{k}}

is a rational number. If

l

is the length of the latus rectum of

\mathrm{H}_{\mathrm{k}}

, then

21 l

is equal to ____________.

Your answerType your answer. Integers and decimals are accepted.