Planar cross-sections of a cone

Victor Hui
The diagram shows a projection of a cone and a cutting plane. The z-x plane is the cutting plane. CG is the symmetry axis of the cone. The vertex of the cone is at C. C lies on the surface of a sphere centred at M on the y-axis. G is the mid-point on the arc AA'. The surface of the sphere contains the circle AA'. To prove and find (a) The position of M fixed the cone angle of the cone. (b) Circulating C on the great circle AGA' yields cuts of all possible elliptical and hyperbolical sections with the same semi-major axis. (c) The conic sections are symmetric about x-axis. (d) Where to place the cone to yield parabolic sections? (e) The locus of O relative to the position of C?