Let H_n: x^2/(1+n)-y^2/(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…

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