If the tangent at a point \(P\) on the parabola \(y^2=3 x\) is parallel to the line \(x+2 y=1\) and the tangents at the …

If the tangent at a point \(P\) on the parabola \(y^2=3 x\) is parallel to the line \(x+2 y=1\) and the tangents at the points \(Q\) and \(R\) on the ellipse \(\frac{x^2}{4}+\frac{y^2}{1}=1\) are perpendicular to the line \(x-y=2\), then the area of the triangle \(P Q R\) is:

[JEE Main 2023, 29 Jan (Shift 2)]