An equilateral triangle $SAB$ is inscribed in the parabola $y^2 = 4ax$ having its focus at '$S$'. If chord $AB$ lies towards the left of $S$, then side length of this triangle is:
Step-by-Step Solution
Key Concept: Use the condition that line $AS$ makes angle $5\pi/6$ to find the parameter value of point $A$.
Let $A = (at_1^2, 2at_1)$ and $B = (at_1^2, -2at_1)$ be points on $y^2 = 4ax$. The slope of $AS$ is $\tan(5\pi/6) = -1/\sqrt{3}$, giving $2at_1/(at_1^2 - a) = -1/\sqrt{3}$. Solving yields $t_1^2 + 2\sqrt{3}t_1 - 1 = 0$, so $t_1 = 2 - \sqrt{3}$ (rejecting the negative solution). Thus $AB = 4at_1 = 4a(2-\sqrt{3})$.
Correct Answer: 2