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Conic Section

Overview

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.

2 figures: one is showing to draw right circular cone and another is double intersected right circular cone with vertex, axis, upper napper, lower nappe, and generator

Take the solid cone and cut them with a sharp knife:

  1. Cut the cone parallel to the base. The section between cone and plane is said to be a circle. cone is cut by knife parallel to base and circle is formed
  2. Cut the cone parallel to slant edge of the cone. The section in between cone and plane is said to be parabola. cone is cut by knife parallel to slant edge of the cone and parabola is formed
  3. 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. cone is cut by knife and ellipse is formed
  4. 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. cone is cut by knife and hyperbola is formed

Exercise

Name the different parts of the cone indicated by a, b, c, d and e. double intersected right circular cone with a, b, c, d and e labels

b = upper nappe
d = generator
e = vertex
a = axis
c = lower nappe
Name the conic section from following figures.
  1. double intersected cone in which lower part is cut by knife in the slightly slanted position
  2. double intersected cone in which upper part is cut by knife in the highly slanted position and lower part is slightly touched
  3. double intersected cone in which lower part is cut by knife in the horizontal position
  4. double intersected cone in which upper part and lower part is cut by knife at once in the slanted position
a = ellipse
b = parabola
c = circle
d = hyperbola
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