Is there an elegant method?

Geometry Level 5

Let the line \(y-\sqrt{3}x+3=0\) intersect the parabola \(2y^{2}=2x+3\) at points \(A\) and \(B\).

Find \(|PA-PB|\), where \(P\) is the point \((\sqrt{3},0)\) and \(PA\) represents the length of line segment ends at \(P\) and \(A\).

\(\)
Notation: \( | \cdot | \) denotes the absolute value function.

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