If eccentricity of the hyperbola `(x^(2))/(cos^(2) theta)-(y^(2))/(sin^(2) theta)=1` is more than `2` when `theta in (0,(pi)/2)`. Find the possible values of length of latus rectum (a) `(3,oo)` (b) `1,3//2)` (c) `(2,3)` (d) `(-3,-2)`

Options

A. `(3,oo)`
B. `(3/2,2]`
C. `(1,3/2]`
D. `(2,3]`

Answer

A

Explanation

`1+tan^(2)theta=e^(2)gt4impliestan^(2)thetagt3impliestheta gt(pi)/3`
Latus rectum length `=(2sin^(2)theta)/(cos theta)=2[sec theta-cos theta]`
As `theta` increases from `(pi)/3` to `(pi)/2`
The length increases from `3 to oo`