Overview of Conic Sections

Overview

This lecture introduces analytic geometry with a focus on the origins, definitions, types, and key properties of conic sections.

Analytic geometry visualizes geometric objects (points, lines, curves, planes) in a coordinate plane using algebraic methods.
Main parts of pre-calculus include analytic geometry, sequences and series, and trigonometry.

Conic sections originated in ancient Greece, linked to mathematicians such as Menaechmus, Euclid, Archimedes, and Apollonius.
Initial names for conic sections were based on the angle of the cone cut: right, obtuse, and acute.
Archimedes included conic ideas in works on geometry, but Apollonius published eight books dedicated to conic sections.
Apollonius established that conic sections are derived from a double-napped cone and named parabola, ellipse, and hyperbola.
Later, Kepler expanded understanding by showing planetary motion (e.g., Mars) follows an elliptical path and identified five conic sections (circle, parabola, ellipse, hyperbola, and lines).

A conic section is a set of points where the distance to a fixed point (focus) and a fixed line (directrix) have a constant ratio (eccentricity).
The general equation for a conic section is: Ax² + Bxy + Cy² + Dx + Ey + F = 0.

Formed when the cutting plane passes through the vertex of the double-napped cone.
Types: point, line, and two intersecting lines.

Formed when the plane does not pass through the cone’s vertex.
Types:
- Circle: cutting plane parallel to the base; eccentricity = 0.
- Parabola: cutting plane parallel to an edge; eccentricity = 1.
- Ellipse: cutting plane not parallel to edge/base; eccentricity < 1.
- Hyperbola: cutting plane intersects both cones; eccentricity > 1.

Analytic Geometry — Study of geometric objects using coordinates and algebra.
Conic Section — Curve formed by intersecting a plane with a double-napped cone.
Degenerate Conic — Special cases like a point, line, or intersecting lines.
Non-Degenerate Conic — Standard curves: circle, parabola, ellipse, and hyperbola.
Focus — Fixed point used in the definition of a conic section.
Directrix — Fixed line used in the definition of a conic section.
Eccentricity — Ratio of the distance from a point to the focus and directrix.