1. Conic Sections
An Introduction

2. Conic Sections - Introduction
A conic is a shape generated by intersecting two lines at a point (vertex) and rotating one line (generator) around the other (axis) while keeping the angle between the lines constant.

3. Conic Sections - Introduction
The resulting collection of points is called a right circular cone
The two parts of the cone are called nappes.
Vertex
Nappe

4. Conic Sections - Introduction
A “conic” or conic section is the intersection of a plane with the cone.
The type of conic is based upon the angle of the cutting plane with the cone, which results in a ratio called eccentricity

5. Conic Sections - Introduction
The plane can intersect the cone at the vertex resulting in a point.
If it goes through the vertex it is a degenerate case (which means we do not care about it very much)
The ones we do care about are the circle, ellipse, parabola, and hyperbola

6. Conic Sections - Introduction
The plane can intersect the cone perpendicular to the axis resulting in a circle.
Eccentricity = 0

7. Conic Sections - Introduction
The plane can intersect one nappe of the cone at an angle to the axis resulting in an ellipse.
0 < Eccentricity < 1

8. Conic Sections - Introduction
The plane can intersect one nappe of the cone at an angle to the axis resulting in a parabola.
Eccentricity = 1

9. Conic Sections - Introduction
The plane can intersect two nappes of the cone resulting in a hyperbola.
Eccentricity > 1

10. Conic Sections - Introduction
GeoGebra demo of eccentricity changing conic
Video of changing cutting plane to generate conic sections
Video of different types of conics

11. Conic Sections - Introduction
The general rectangular equation for any conic is Ax2 + Bxy + Cy2 + Dx + Ey + F = 0 where A, B, C, D, E, and F are all numbers and A & C are not both zero
The Bxy term would make the conic rotated and not “square” with the axis, so we will not cover this