Now a whole family circling a parabola!

If the locus of centers of a family of circles which pass through the vertex of the parabola $y^2=4ax$ and cut the parabola orthogonally at the other point of intersection can be represented as

$\alpha y^2 \left( \alpha y^2 + x^2 - \beta ax \right) = ax {\left( \gamma x - \delta a \right)}^2$

where $\alpha \ , \beta \ , \gamma \ , \delta$ are positive integers.

Find the value of $\alpha + \beta + \gamma + \delta$ .