add a conic branch tangent to a spline

I need to add a conic branch that is tangent to a spline. The result looks good, but zooming shows otherwise.

Your help would be appreciated!

Fig.1 The spline (red). ___________________________________________________________________    Fig.2 The conic branch (red) is meant to be tangent to the spline. The grey phantom line is used as contruction line, that is tangent to the spline and holds the conic anchor.Fig.3 Unfortunately the "Point on Curve" contraint is lost!Fig.4 Deleting the contruction line would restore the "Point on Curve"" constraint.

I'm not sure whether so frequent posting would make me unwelcom in this forum. If this is true, please  give me a notice. Thank you!