Conic Sections

Intuition and core definition

Conic sections are curves obtained by intersecting a double cone with a plane: circles, ellipses, parabolas, and hyperbolas. They can also be defined as loci using distances to foci and directrices. The plane’s angle and position determine which curve appears.

Notation, language, and conditions

In general quadratic form $Ax^2+Bxy+Cy^2+Dx+Ey+F=0$, the discriminant $B^2-4AC$ helps classify nondegenerate conics after accounting for rotation: negative suggests ellipse/circle, zero parabola, positive hyperbola. Standard forms expose centres, vertices, axes, and parameters more directly.

Why this idea matters

Conic classification connects the geometry of slicing a cone with algebraic patterns in quadratic equations.

A dependable method

1. Identify whether the information is geometric, locus-based, or algebraic.
2. For an equation, group quadratic and linear terms and inspect signs and coefficients.
3. Complete squares and rotate axes if an $xy$ term requires it at the studied level.
4. Match to a standard form and read defining features.
5. Check the equation with vertices, symmetry, and domain behaviour.

Worked example

Common mistakes