Conic sections are curves formed by the intersection of a plane with a double cone. Depending on the angle of the cut, you get a circle, ellipse, parabola, or hyperbola. These shapes appear everywhere — from planetary orbits to satellite dishes.
Parabola
A parabola is the set of all points equidistant from a fixed point (focus) and a fixed line (directrix). Standard form: y = ax² (vertical) or x = ay² (horizontal). Key features: vertex, focus, directrix, axis of symmetry.
Ellipse
An ellipse is the set of all points where the sum of distances to two fixed points (foci) is constant. Standard form: x²/a² + y²/b² = 1. Key features: semi-major axis a, semi-minor axis b, two foci at (±c, 0) where c² = a² − b².
Hyperbola
A hyperbola is the set of all points where the difference of distances to two foci is constant. Standard form: x²/a² − y²/b² = 1. Key features: two branches, two foci, asymptotes y = ±(b/a)x.