The size of the parameter set in the two representations
of either 5 (2 center coordinates, 2 radii, 1 alignment angle)
or 6 (quadratic form $Ax^2+Bxy+Cy^2+Dx+Ey+F=0$) is
due to the fact that the algebraic equation/form is invariant
to the space of solutions (x,y) if it is divided through any
of the 6 (nonzero coefficients): $x^2+(B/A)*x*y+(C/A)*y^2+(D/A)*x +(E/A)*y+F/A=0$.
So the quadratic form has effectively only 5 independent parameters.