On a rectangular hyperbola `x^(2)-y^(2)=a^(2),a gt 0`, three points A,B,C are taken as follows : A = (-a,0): B and C are placed symmetrically with respect to the x-axis on the branch of the hyperbola not containing A suppose that the triangle ABC is equilateral. If the side-length of the triangle ABC is ka,then k lies in the interval

Options

A. (0,2]
B. (2,4]
C. (4,6]
D. (6,8]

Answer

B

Explanation

`x^(2)-y^(2)=a^(2)`
A(-a,0)
`B(a sec theta, a tan theta)`
`C(a sec theta, -a tan theta)`
`M_(AB)= tan 30^(@)=(a tan theta)/(a sin theta +a)=(1)/(sqrt(3))`
`sqrt(3) tan theta =1+ sin theta`
`sqrt(3) sin theta =1+ sec theta`
`(sqrt(3) tan theta-1)^(2)= sec^(2) theta`
`3 tan^(2) theta =2 sqrt(3) tan theta +1=1+ tan^(2) theta`
`2 tan^(2) theta - 2 sqrt(3) tan theta=0`
`tan theta=sqrt(3)`
Side length = `2a tan theta`
`=2a sqrt(3)`
`=2sqrt(3)a`
`K=2sqrt(3)`