Conic Sections Summary

Algebra 2 · Axiom Academy

How slicing a cone at different angles produces four curves, each with its own equation, foci, and defining property. All four conics — circle, ellipse, parabola, hyperbola — come from the same operation: slicing a plane through a cone. Every conic has a focal (distance-based) definition, and that definition is exactly what generates its algebraic equation. Ellipses are built on a sum of two focal distances; hyperbolas are built on a difference — that sign flip changes everything downstream. The a,b,c relationship flips to match: c^2=a^2-b^2 for ellipses, c^2=a^2+b^2 for hyperbolas. The discriminant B^2-4AC identifies any conic straight from its general-form equation — no graphing required. Core Concept One Shape, Four Curves Every conic section comes from slicing a double cone with a flat plane. The angle of the cut determines which curve appears. Horizontal slice: a circle (a special-case ellipse). Tilted slice through one nappe: an ellipse. Slice parallel to the cone's side: a parabola. Vertical slice through both nappes: a hyperbola. A circle is the set of all points at a fixed distance r (the radius) from a center point (h,k) — the special case of an ellipse where both foci merge into one. When to use: any "fixed distance from a point" problem — Circle Designer, GPS range rings, circle-and-line intersections. Watch out for: a general-form equation ( x^2+y^2+Dx+Ey+F=0 ) has to be completed to the square before you can read off the center and radius.

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