Line Rotation and Slope (Jee Mains)

Equations of a Line

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Line Rotation and Slope

A line passing through the point P(a, θ) makes an acute angle α 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 3a²tan²α – 2 $\sqrt 3$ is

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