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 length of the latus rectum of H k , then 21 l is equal to

Step-by-step solution

Given, Equation of hyperbola, H n : x 2 1 + n - y 2 3 + n = 1 ,   n ∈ ℕ Now using the eccentricity formula we get, e 2 = 1 + 3 + n 1 + n ⇒ e 2 = 2 n + 4 n + 1 = 2 ( n + 2 ) n + 1 Now check when ( n + 1 ) = 9 , 25 , 49 , … … is perfect square and then at the same time 2 n + 2 should also be perfect square, So checking for n = 8 →   e 2 = 20 9 n = 24 →   e 2 = 52 25 n = 48 → e 2 = 100 49 ⇒ e = 10 7 Hence, for n = 48 both n + 1   &   2