What plane shapes may you get if you cut a carrot (right circular cone) with a knife at different angles?
Let AB and CD be two fixed lines intersecting at the point O at an angle (0° < θ < 90°). If the plane containing these two lines AB and CD is rotated about the line CD, then the surface generated by AB is called the right circular cone. O is the vertex, the CD is the vertical axis, θ is the vertical angle and AB is the generator of the cone.
Take the solid cone and cut them with a sharp knife:
- Cut the cone parallel to the base. The section between cone and plane is said to be a circle.
- Cut the cone parallel to slant edge of the cone. The section in between cone and plane is said to be parabola.
- Cut the cone in such a way that the angle between plane and axis of cone is greater than vertical angle of cone. But less than 90°. the section in between cone and the plane is said to be ellipse.
- Intersect double circular cone parallel to edge of the cone (slant edge). The pair of parabola or the sections formed between double cone and plane is said to be hyperbola.
Name the different parts of the cone indicated by a, b, c, d and e.
d = generator
e = vertex
a = axis
c = lower nappe
b = parabola
c = circle
d = hyperbola