Conic Sections 11.1 - An Introduction

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.

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

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

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

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

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

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

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

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

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