MathFormaCurves of the Second Order
Study axis-aligned second-order curves from their general equations, including classification by the sign of AC, reduction to canonical form, and finding the centre or vertex, foci, eccentricity, directrices, and asymptotes.
0/5SOLVED
BASIC DEFINITIONS
Basic Definitions
The definitions, canonical equations, parameters, and construction rules for the ellipse, hyperbola, and parabola: the tools used throughout the problems below.
1Problem 1 · Translated EllipseReduce a general second-order equation to canonical form, then find the centre, semi-axes, foci, directrices and eccentricity.Start →2Problem 2 · Degenerate Conic: Single PointComplete the square on a second-order equation and recognise when the locus collapses to a single point.3Problem 3 · Imaginary Ellipse (No Real Points)Complete the square on a second-order equation and recognise when it has no real points.4Problem 4 · Translated HyperbolaReduce a general second-order equation to canonical form, then find the centre, axes, foci, asymptotes, directrices and eccentricity.5Problem 5 · Degenerate Conic: Two Intersecting LinesComplete the square on a second-order equation and recognise when it factors into a pair of intersecting lines.