Uniform Zariski's theorem on fundamental groups

The Zariski theorem says that for every hypersurface in a complex projectiv e (resp. affine) space and for every generic plane in the projective (resp. affine) space the natural embedding generates an isomorphism of the fundam ental groups of the complements to the hypersurface in the plane and in the space. If a family of hypersurfaces depends algebraically on parameters th en it is not true in general that there exists a plane such that the natura l embedding generates an isomorphism of the fundamental groups of the compl ements to each hypersurface from this family in the plane and in the space. But we show that in the affine case such a plane exists after a polynomial coordinate substitution.