A line passing through the point \( P(a, 0) \) makes an acute angle \( \alpha \) with the positive \( x \)-axis. Let this line be rotated about the point \( P \) through an angle \( \dfrac{\alpha}{2} \) in the clock-wise direction. If in the new position, the slope of the line is \( 2 - \sqrt{3} \) and its distance from the origin is \( \dfrac{1}{\sqrt{2}} \), then the value of \( 3a^2 \tan^2 \alpha - 2\sqrt{3} \) is:
