Let AT a common tangent, touching the circle with center O in A.
Let M the perpendicular projection of focus F on this tangent, therefore on the tangent at apex of parabola.
Consider point A' symmetric of A with respect to M, and the circle tangent in A' to AT, and centered in O' on the parabola axis
Through projection perpendicular to AT, F is midpoint of OO'
Also, the tangent at apex, being the perpendicular to axis, and going through the midpoint of the common tangent AA' to the circles is therefore the radical axis of these circles.
The common tangents to a parabola with focus F and to a circle centered in O on the axis
are tangent to a second circle centered in O' symmetric of O with respect to focus F,
and such that the tangent at apex of parabola is the radical axis of these circles.
In the applet, the draggable blue points F, S, O, r define the parabola and the given circle.
This gives the wanted construction :
Depending on the circle position, there are 0, 2 or 4 common tangents, may be collapsing by pairs.
If the given circle is centered in F, the construction fails : the two circles (O) and (O') overlap.
But in the limit, the midpoint of their common tangent becomes the intersection of the given circle with the radical axis.
The common tangent is then the tangent to (O) in the intersection point of this circle and the tangent to apex.
The applet doesn't handle this specific case.
When (O) is tangent to parabola, the circle (O') degenerates to a point, the applet doesn't handle this case either.
(The common tangent is the tangent from O')
When (O) is tangent to parabola in apex S, the common tangent in S is not constructed either (because of the method used here to construct common tangents to two circles).