A line passing through the point P(a, 0) makes an acute angle α with the positive x -axis. Let this line be rotated about the point P through an…

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