Circle

If $\left(0, \frac{3}{4}\right)$ is the radical centre of the circles $S \equiv x^2+y^2+\alpha x+6 y=0, S \equiv x^2+y^2+2 \alpha x+\alpha y+6=0$ and $S^{\prime \prime} \equiv x^2+y^2+6 \alpha x-\alpha y+3=0$, then the distance between the radical centre and the centre of the circle $S^{\prime}=0$ is