A line passing through the point \(\mathrm{P}(\mathrm{a},0)\)makes an acute angle \(\alpha\) with the positive x -axis. …

A line passing through the point \(\mathrm{P}(\mathrm{a},0)\)makes an acute angle \(\alpha\) with the positive x -axis. Let this line be rotated about the point P through an angle \(\frac{\alpha }{2}\) 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 \(\frac{1}{\sqrt{2}}\), then the value of \(3{\mathrm{a}}^{2}{\tan }^{2}\alpha -2\sqrt{3}\) is

[JEE Main 2025, 8 Apr (Shift 1)]