If LM is a double ordinate of the hyperbola `(x^(2))/(a^(2)) + (y^(2))/(b^(2)) =1` such that OLM is an equilateral triangle, O being the centre of the hyperbola, then the eccentricity e of the hyperbola. Satisfies

Options

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

Answer

D

Explanation

`(x^(2))/(a^(2)) - (y^(2))/(b^(2)) = 1`
`rArr x^(2) = (b^(2) + y^(2))(a^(2))/(b^(2))`…….(i)
Also,
`x^(2) + l^(2) = 4l^(2)`
`x^(2) = 3l^(2)"........"(ii)`
`(i)` & `(ii) rArr (a^(2)(b^(2) + l^(2)))/(b^(2)) = 3l^(2)`
`rArr a^(2)b^(2) + a^(2)l^(2) = 3b^(2)l^(2)`
`rArr l^(2) = (a^(2)b^(2))/(3b^(2) - a^(2)) gt 0`
`rArr 3b^(2) - a^(2) gt 0 rArr (b^(2))/(a^(2)) gt (1)/(3)`
`:. 1 + (b^(2))/(a^(2)) gt 1 + (1)/(3)`
`e^(2) gt (4)/(3) rArr e gt (2)/(sqrt(3))`