The eccentricity of the hyperbola whose length of the latus rectum is equal to 8 and the length of its conjugate axis is equal to half of the distance between its foci, is : (1) `4/3` (2) `4/(sqrt(3))` (3) `2/(sqrt(3))` (4) `sqrt(3)`

Options

A. `3//4`
B. `4//sqrt3`
C. `2//sqrt3`
D. none of these

Answer

C

Explanation

We have
`(2b^(2))/(a)=8`
`and 2b=(1)/(2)(2ae)`
`therefore" "(2)/(a)((ae)/(2))^(2)=8`
`"or "ae^(2)=16" (1)"`
Also, `2xx(b^(2))/(a)=8`
`"or "b^(2)=4a`
`"or "a^(2)(e^(2)-1)=4a`
`"or "ae^(2)-a=4" (2)"`
From (1) and (2), we have
`16-(16)/(e^(2))=4`
`"or "(16)/(e^(2))=12`
`"or "e=(2)/(sqrt3)`