Equation of circle touching a parabola Suppose we have a parabola $y^2=4x$  . Now, how to write equation of circle touching parabola at $(4,4)$ and passing thru focus?
I know that for this parabola focus will lie at $(1,0)$ so we may assume general equation of circle and satisfy the points in it . hence
$$x^2 +y^2 + 2gx + 2fy +c =0$$ should be equation of general circle but on satisfying given points we get 2 equations and 3 variables . I think I am missing something, what to do?

The center lies at the intersection of the perpendicular to the tangent and the bisector of the two known points.
$$2(x-4)+(y-4)=0$$
$$(4-1)(x-\frac{4+1}2)+(4-0)(y-\frac{4+0}2)=0.$$
Hence
$$x=\frac{13}2,y=-1.$$
