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There is another way to form Conics.           

Conics (1)

There are Seven Conics
(Conic Sections)
Made by Cutting a Cone with a Plane
Circle
Ellipse
Point
Line-Pair
Hyperbola
Parabola
Line

The first thing to note is that
there is no way for a cutting plane to avoid cutting the cone,
so a conic of some kind is inevitable,

but


- if the cutting plane's attitude with respect to the cone is such
that it cuts a plane at right angles to the cone's axis in a line
that is partly in the blue enclosure,

then the conic must be a

parabola, (plane misses the vertex, and lies parallel to a line of the cone.)
hyperbola, (plane misses the vertex, and does not lie parallel to a line of the cone.)
line-pair, (plane cuts the vertex, and does not lie parallel to a line of the cone.)
line. ( plane cuts the vertex, and lies parallel to, and in, a line of the cone.)

 

- if the cutting plane's attitude with respect to the cone is such
that it cuts a plane at right angles to the cone's axis in a line
that is nowhere in the blue enclosure,

then the conic must be a

circle, (the plane misses the vertex, and is at right angles to the axis.)
ellipse, (the plane misses the vertex, and is not at right angles to the axis.)
point. (the plane cuts the vertex, and may be at right angles to the axis.)

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