If PQ is a double ordinate of the hyperbola `(x^(2))/(a^(2))-(y^(2))/(b^(2))=1`
such that OPQ is an equilateral triangle, O being the centre of the hyperbola, then find range of the eccentricity (e) of the hyperbola.

Options

A. `1 lt e lt (2)/(sqrt(3))`
B. `e=(2)/(sqrt(3))`
C. `e=(sqrt(3))/(2)`
D. `e gt (2)/(sqrt(3))`

Explanation

Let the length of each side of the equilateral triangle `OPQ` be `l` units. Then, the coordinates of `P` are `((sqrt(3l))/(2),(l)/(2))`.
Point `P((sqrt(3l))/(2),(l)/(2))` lies on the hyperbola `(x^(2))/(a^(2))-(y^(2))/(b^(2))=1`

`:. (3l^(2))/(4a^(2))-(l^(2))/(4b^(2))=1`
`implies(3b^(2)-a^(2))l^(2)=4a^(2)b^(2)`
`implies(3e^(2)-4)l^(2)=4a^(2)(e^(2)-1)impliesl=2asqrt((e^(2)-1)/(3e^(2)-4))`
Since `l` is real and `e gt 1`.
`:.3e^(2)-4 gt 0impliese^(2) gt (4)/(3)impliese gt (2)/(sqrt(3))`