Conic Sections
General Equation
Ax²+Bxy+Cy²+Dx+Ey+F=0. Discriminant B²−4AC: <0 ellipse, =0 parabola, >0 hyperbola.
Examples
Example 1. Classify x²+y²=9.
Solution. Circle (A=C, B=0).
In Depth
Conic sections are the curves formed by intersecting a double cone with a plane. Depending on the angle: a horizontal cut gives a circle, a tilted cut gives an ellipse, a cut parallel to the side gives a parabola, and a steeper cut gives a hyperbola. A degenerate cut through the apex gives a point, line, or pair of lines.
The unified focus-directrix definition: a conic is the locus of points where the ratio of distance to a focus and distance to a directrix equals the eccentricity \(e\). Circle: \(e=0\). Ellipse: \(0
In polar coordinates with focus at origin: \(r = \ell/(1-e\cos\theta)\) where \(\ell\) is the semi-latus rectum. This single formula describes all conics and is used in orbital mechanics — Kepler's first law states that planetary orbits are conics with the Sun at a focus.
The general second-degree equation \(Ax^2+Bxy+Cy^2+Dx+Ey+F=0\) represents a conic. The discriminant \(\Delta=B^2-4AC\) classifies it: \(\Delta<0\) (ellipse/circle), \(\Delta=0\) (parabola), \(\Delta>0\) (hyperbola). Rotation by angle \(\theta=\frac{1}{2}\arctan(B/(A-C))\) eliminates the \(xy\) term.
Reflective properties make conics useful in optics and engineering: parabolic mirrors focus parallel rays to a point (telescopes, satellite dishes); elliptical mirrors reflect from one focus to the other (whispering galleries, lithotripsy); hyperbolic mirrors appear in Cassegrain telescopes.
Key Properties & Applications
Kepler's first law: planetary orbits are ellipses with the Sun at one focus. The eccentricity of Earth's orbit is 0.017 (nearly circular); Pluto's is 0.25. Comets often have highly eccentric elliptical or parabolic orbits.
Parabolic reflectors focus parallel rays to a point (the focus): satellite dishes, telescope mirrors, and car headlights use parabolic shapes. The reflective property — a ray from the focus reflects parallel to the axis — follows from the equal-angle reflection law and the focus-directrix definition.
Hyperbolas appear in navigation (LORAN uses hyperbolic position lines), in the Bohr model of atomic scattering (Rutherford scattering), and in the shape of cooling towers. The asymptotes of a hyperbola \(x^2/a^2-y^2/b^2=1\) are \(y=\pm(b/a)x\).
Further Reading & Context
The study of conic sections connects to many areas of mathematics and its applications. Understanding the foundational definitions and theorems provides the basis for advanced work in analysis, algebra, and applied mathematics.
Historical development: most mathematical concepts evolved over centuries, with contributions from mathematicians across many cultures. The modern axiomatic treatment provides rigor, while computational tools enable practical application.
In modern mathematics, this topic appears in graduate courses and research across pure and applied mathematics. Connections to computer science, physics, and engineering make it a versatile and important area of study. Mastery of the core results and techniques opens doors to research in number theory, analysis, geometry, and beyond.
Recommended next steps: work through the standard theorems with full proofs, explore the connections to related topics listed above, and practice with a variety of problems ranging from computational exercises to theoretical proofs. The interplay between different areas of mathematics is one of the subject's greatest rewards.
Deep Dive: Conic Sections
This lesson extends core ideas for conic sections with rigorous reasoning, edge-case checks, and application framing in geometry.
Practice Set
Practice. Derive one main result on this page and validate with a numeric or geometric check.
Goal. Confirm assumptions, transformation steps, and final interpretation.
References & Editorial Notes
- Stewart, Calculus.
- Strang, Introduction to Linear Algebra.
- Apostol, Mathematical Analysis.
Last editorial review: 2026-04-14.