Let the focal chord \( PQ \) of the parabola \( y^{2} = 4x \) make an angle of \( 60^\circ \) with the positive \( x \)-axis, where \( P \) lies in the first quadrant. If the circle, whose one diameter is \( PS \), \( S \) being the focus of the parabola, touches the \( y \)-axis at the point \( (0, \alpha) \), then \( 5\alpha^{2} \) is equal to:
