Hello, George!

If no one has ever explained Conic Section (and conic curves) to you,

. . no wonder that you don't have a clue . . .

The vertex angle of 105° is very inconsiderate.

I'll change it to 120°.

The vertex angle of a double napped cone is 105°.

The angle $\displaystyle \theta$ is the angle measured from the axis to the cutting plane.

For what angle $\displaystyle \theta$, or range of values for $\displaystyle \theta$, does the cutting plane need to

intersect the conical surface in order to generate each of the following conics?

(a) hyperbola . . (b) Ellipse . . (c) circle . . (d) parabola

If the plane is perpendicular to the axis of the cone (θ = 90°),

the intersection is a circle. Code:

*
/:\
/ : \
/ : \
/60°: \
/ : \
o o o o o o o o o
/ θ: \
/ : \

If the cutting plane is __parallel__ to the slant of the cone (θ = 30°)

. . the intersection is a parabola. Code:

*
/:\
/ : \o
/ : o\
/ :o \
/ o \
/ o: \
/ o : \
/ o θ| \

In between those two angles (30° < θ < 90°)

. . the intersection is an ellipse. Code:

*
/:\
/ : \
/ : \ o
/ : \ o
/ : o \
/ o \
/ o : \
/ o θ : \
o : \

If the plane is steeper than that of the parabola (θ < 30°),

. . the intersection is a hyperbola.

Note that the plane cuts both nappes of the cone. Code:

\ : o /
\ : /
\ : o/
\ : /
\ : o
\ : /
\ : /o
\:/
* o
/:\
/ :o\
/ : \
/ o \
/ : \
/ o: \
/ : \
/ o : \