An equilateral triangle `S A B` is inscribed in the parabola `y^2=4a x` having its focus at `Sdot` If chord `A B` lies towards the left of `S ,` then the side length of this triangle is (a)`2a(2-sqrt(3))` (b) `4a(2-sqrt(3))` (c)`a(2-sqrt(3))` (d) `8a(2-sqrt(3))`

Options

A. `2a(2-sqrt(3))`
B. `4a(2-sqrt(3))`
C. `a(2-sqrt(3))`
D. `8a(2-sqrt(3))`

Answer

B

Explanation

(2) Let `A-=(at_(1)^(2),2at_(1)),B=-(at_(2)^(2),-2at_(1))`.
We have
`m_(AS)=tan((5pi)/(6))`
`or(2at_(1))/(at_(1)^(2)-a)=-(1)/(sqrt(3))`
`ort_(1)^(2)+2sqrt(3)t_(1)-1=0`
`ort_(1)=-sqrt(3)pm2`
Clearly, `t_(1)=-sqrt(3)-2` is rejected.
Thus, `t_(1)=(2-sqrt(3))`.
Hence, `AB=4at_(1)=4a(2-sqrt(3))`.