Conic Sections — Mathematics Knowledge System

Overview

A conic section (or simply a conic) is a curve obtained by intersecting a plane with the surface of a double cone. The three fundamental types are the ellipse, the parabola, and the hyperbola; the circle is a special case of the ellipse (sometimes historically counted as a fourth type). Ancient Greek mathematicians studied these curves extensively, culminating around 200 BCE in Apollonius of Perga's systematic treatise on their properties. Every conic can also be described algebraically as the solution set of a general second-degree (quadratic) equation in two variables.

Intuition

Picture two ice-cream-cone shapes joined tip-to-tip, extending infinitely in both directions, and slice through them with a flat plane. Slice perpendicular to the axis and you get a circle. Tilt the plane a little and the circle stretches into an ellipse. Tilt it until the plane is parallel to the cone's slanted side and the curve opens up forever — a parabola. Tilt it even further so the plane cuts through both nappes of the double cone, and you get two separate curves opening away from each other — a hyperbola. The steepness of the cutting plane relative to the cone's side is exactly what the eccentricity measures algebraically.

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Formal Definition

Definition

Algebraically, every conic section is the solution set of a general second-degree equation in two variables:

Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0

A, B, C are not all zero

General conic equation

\Delta = B^2 - 4AC

Classifies the type of conic when the curve is non-degenerate

Discriminant

\Delta < 0 \implies \text{ellipse (or circle if } A=C,\ B=0\text{)}

Negative discriminant

\Delta = 0 \implies \text{parabola}

Zero discriminant

\Delta > 0 \implies \text{hyperbola}

Positive discriminant

e = \frac{c}{a}, \qquad \ell = pe,\qquad p + c = \frac{a}{e}

Eccentricity e relates a conic's focus-directrix distance p, its linear eccentricity c, and semi-major axis a; e=0 is a circle, 0<e<1 an ellipse, e=1 a parabola, e>1 a hyperbola

Notation

Notation	Meaning
$e$	Eccentricity — a single number classifying the conic's shape (0 = circle, 0<e<1 ellipse, e=1 parabola, e>1 hyperbola)
$F$	A focus: a fixed point used in the geometric (focus/directrix) definition of a conic
$a, b, c$	Semi-major axis, semi-minor axis, and linear eccentricity (focal distance from center)
$\Delta = B^2-4AC$	Discriminant of the general second-degree equation, determining the conic's type

Derivation

Every conic (other than a degenerate case) can be defined uniformly using a focus F, a directrix line, and an eccentricity e ≥ 0: it is the set of points P whose distance to F is e times the distance to the directrix.

|PF| = e \cdot d(P, \text{directrix})

Unified focus-directrix definition of a conic

e = 0 \implies \text{circle (degenerate case: directrix at infinity)}

0 < e < 1 \implies \text{ellipse}

e = 1 \implies \text{parabola}

e > 1 \implies \text{hyperbola}

\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \quad (a,b>0)

Placing the focus/directrix pair appropriately and simplifying recovers the familiar standard-form equation, here for the ellipse

Properties

Unified quadratic form

\text{Every conic satisfies } Ax^2+Bxy+Cy^2+Dx+Ey+F=0

Discriminant classification

B^2-4AC < 0 \Rightarrow \text{ellipse},\ =0 \Rightarrow \text{parabola},\ >0 \Rightarrow \text{hyperbola}

Eccentricity range

e=0 \text{ (circle)},\ 0<e<1 \text{ (ellipse)},\ e=1 \text{ (parabola)},\ e>1 \text{ (hyperbola)}

Reflective property

\text{Each conic type has a distinct optical reflective property exploited in mirrors, antennas, and orbits}

Applications

Kepler's laws show that orbits of planets and comets under gravity are conic sections: ellipses for bound orbits, parabolas and hyperbolas for unbound trajectories.

Worked Examples

Identify A=3, B=0, C=3 and compute the discriminant.
$\Delta = B^2 - 4AC = 0 - 4(3)(3) = -36 < 0$
A negative discriminant with A=C and B=0 signals a circle (a special ellipse).
$\Delta < 0,\ A = C \Rightarrow \text{circle}$

Answer: The equation represents a circle (a degenerate case of the ellipse, since A = C and B = 0).

Practice Problems

Difficulty 5/10

Classify the conic 5x² - 2xy + 5y² - 4 = 0 using the discriminant.

Difficulty 5/10

Classify the conic x² - y² - 2x + 4y - 3 = 0 using the discriminant.

Difficulty 5/10

A parabolic arch bridge is 40 m wide at the base and 10 m high at the center. With the vertex at the top and origin at the center, its equation is y = 10 − (10/400)x². What is the height 10 m from the center?

Common Mistakes

Common Mistake

Assuming any second-degree equation Ax²+Bxy+Cy²+Dx+Ey+F=0 automatically produces a 'nice' curve.

Degenerate cases exist — the equation can describe a single point, a line, two intersecting lines, or the empty set, depending on the linear terms and constant. The discriminant classifies the type only among non-degenerate conics.

Common Mistake

Confusing eccentricity with the linear eccentricity c or with the semi-axes a, b directly.

Eccentricity e = c/a is a dimensionless ratio (a pure number describing shape), while a, b, c carry actual length/distance information about a specific conic.

Quiz

The discriminant B² − 4AC of a general conic is NEGATIVE. The conic is:

a parabolaan ellipse (or circle)a hyperbolaa straight line

Which structure is naturally modelled by a conic section?

A parabolic arch bridge or satellite dishA spreadsheetA hash tableA sorting network

Historical Background

Conic sections were first studied geometrically — as literal cross-sections of a cone cut by a plane — by Greek mathematicians such as Menaechmus in the 4th century BCE, reportedly in the course of trying to solve the problem of doubling the cube. Apollonius of Perga's treatise 'Conics' (c. 200 BCE) gave the first exhaustive, systematic treatment, introducing the names 'ellipse,' 'parabola,' and 'hyperbola' still used today. The algebraic unification of all three curves under a single second-degree equation came much later, after Descartes and Fermat founded coordinate geometry in the 17th century.

c. 350 BCE
Menaechmus studies curves formed by cutting a cone with a plane
Menaechmus
c. 200 BCE
Apollonius of Perga writes 'Conics,' systematizing the theory and naming the three curves
Apollonius of Perga
17th century CE
Descartes and Fermat's coordinate geometry allows conics to be classified by algebraic equation

Summary

A conic section is the curve formed by a plane slicing a double cone: circle, ellipse, parabola, or hyperbola depending on the angle of the cut.
Every conic satisfies the general second-degree equation Ax²+Bxy+Cy²+Dx+Ey+F=0.
The discriminant Δ=B²-4AC classifies the (non-degenerate) type: negative→ellipse, zero→parabola, positive→hyperbola.
The unified focus-directrix definition |PF| = e·d(P, directrix) ties all conics together via the single eccentricity parameter e.
Conics were studied geometrically by the ancient Greeks (Apollonius, c. 200 BCE) long before Descartes' 17th-century algebraic framework unified them.

References

WebsiteWikipedia — Conic section
Historical sourceApollonius of Perga, Conics (c. 200 BCE)