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?