The correct option is D (3,∞)
Here, a2=cos2θ and b2=sin2θ
Eccentricity, e=√1+(ba)2=√1+tan2θ=secθ
Given, e>2∴secθ>2
⇒cosθ<12⇒θ∈(π3,π2)
Let L be the length of the latus rectum.
Then L=2b2a=2×sin2θcosθ=2tanθ.sinθ, which is strictly increasing in θ∈(π3,π2).
So, minimum and maximum value of latus rectum occurs at θ=π3 and θ=π2 respectively.
Lmin=2×sin2π3cosπ3=3
Lmax=2×sin2π2cosπ2→∞
∴L∈(3,∞)