One might consider working a slightly simpler example, first.

Rather than (10,12,15), why not try it at (0,0,21.65640783).

With a 10º vertex angle, this gives a nice cone:

With your 6-radius sphere, this is easily solved for the intersection, giving first z = 5.838274311 and defining the circle nicely with radius 1.38367376.

Your remaining task is to rotate the result to where you want it. Alternatively, one could rotate where you want it to the easier solution.