Conic Sections — Parabola, Ellipse, Hyperbola, Circle

A conic section is the intersection of a plane with a right circular double cone; the angle of the cut determines whether you get a circle, ellipse, parabola, or hyperbola. Algebraically, every conic is the graph of a second-degree equation $A{x}^2+Bxy+C{y}^2+Dx+Ey+F=0$, making conics the simplest curves beyond straight lines. They matter because they describe an astonishing range of physical phenomena: orbits under gravity, the focusing of light and sound, and the trajectories of charged particles. A surprising fact: all four conics share a single unified description via the eccentricity $e$ — a single number that smoothly morphs a circle ($e=0$) into an ellipse, then a parabola ($e=1$), then a hyperbola ($e>1$).

The Core Insight

Every conic is a locus — a set of points satisfying a distance condition. The deepest unifying idea is the focus–directrix property: for a fixed point $F$ (focus), a fixed line $D$ (directrix), and a fixed positive number $e$ (eccentricity),

The geometry of the cone and the algebra of the quadratic equation are two faces of this one locus condition.

Formal Statement

The Four Conics as Loci

Let distances be measured in the Euclidean plane.

• Circle: the locus of points at fixed distance $r$ (radius) from a fixed center $C$.
• Ellipse: the locus of points $P$ for which the sum of distances to two foci $F_1,F_2$ is a constant $2a$:

• Parabola: the locus of points equidistant from a focus $F$ and a directrix line $D$:

• Hyperbola: the locus of points $P$ for which the absolute difference of distances to two foci is a constant $2a$:

Standard-Form Equations

With axes aligned to the coordinate axes and centers/vertices placed conveniently:

Here $a$ = semi-major (or transverse) axis, $b$ = semi-minor (or conjugate) axis, $c$ = focal distance from center, $p$ = focal parameter, $r$ = radius.

General Second-Degree Equation & the Discriminant

Every conic is described by

The type is classified by the discriminant $\Delta =B^2-4AC$:

Why It Holds / Derivation

Deriving the Ellipse Equation from the String Property

We derive equation [6] from the locus definition [2].

1. Place foci at $F_1=(-c,0)$ and $F_2=(c,0)$. Let $P=(x,y)$.
2. The locus condition is $\sqrt{(x+c{)}^2+y^2}+\sqrt{(x-c{)}^2+y^2}=2a$.
3. Isolate one radical: $\sqrt{(x+c{)}^2+y^2}=2a-\sqrt{(x-c{)}^2+y^2}$.
4. Square both sides:

1. Expand and cancel: $4cx=4a^2-4a\sqrt{(x-c{)}^2+y^2}$, so $\sqrt{(x-c{)}^2+y^2}=a-\frac{c}{a}x$.
2. Square again: $(x-c{)}^2+y^2=a^2-2cx+\frac{c^2}{a^2}{x}^2$.
3. Expand left side and simplify: $x^2\left(1-\frac{c^2}{a^2}\right)+y^2=a^2-c^2$.
4. Let $b^2=a^2-c^2$. Then $x^2\frac{b^2}{a^2}+y^2=b^2$.
5. Divide by $b^2$: $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1.$ $■$

Eccentricity Unifies All Conics

Starting from the focus–directrix ratio [1] with focus $F=(c,0)$, directrix $x=c/e^2$ (one valid placement), and ratio $e$:

1. The condition $|PF|=e\cdot \text{dist}(P,D)$ becomes $\sqrt{(x-c{)}^2+y^2}=e\left|x-\frac{c}{e^2}\right|$.
2. Square: $(x-c{)}^2+y^2=e^2{x}^2-2cx+\frac{c^2}{e^2}$.
3. Collect: $(1-e^2)x^2+y^2=\frac{c^2}{e^2}(1-e^2)$ (after algebra).
4. If $0<e<1$: $1-e^2>0$ → divide to get an ellipse form.
5. If $e=1$: the $x^2$ terms cancel, leaving $y^2=\text{linear in }x$ → a parabola.
6. If $e>1$: $1-e^2<0$ → sign flip yields a hyperbola.
7. As $e\to 0$ the directrix recedes to infinity and the ellipse becomes a circle. $■$

The four conic sections as plane cuts of a double cone, with eccentricity increasing left to right

Apollonius of Perga | c. 262 – c. 190 BCE

Apollonius, a Greek geometer working in Alexandria and Pergamon, wrote the eight-book treatise Conics — the definitive ancient work on the subject. He coined the names ellipse ("falling short"), parabola ("application/equal"), and hyperbola ("exceeding"), borrowing terms from the Pythagorean theory of "application of areas." Crucially, he showed all three curves arise from a single cone by varying only the cutting angle, rather than requiring three different cones as earlier geometers believed. His work, deeply geometric and without algebra, anticipated coordinate ideas by nearly 2000 years and directly informed Kepler and Newton.

Building Intuition

Think of a flashlight beam (a cone of light) hitting a flat wall.

• Point it straight on → circular spot.
• Tilt it slightly → the spot stretches into an ellipse.
• Tilt until one edge of the cone runs parallel to the wall → the spot opens into an unbounded parabola.
• Tilt further, so the wall cuts both the upward and downward parts of the cone → two separate branches, a hyperbola.

The eccentricity measures "how stretched": $e=0$ is a perfect circle, and as $e$ climbs toward $1$ the ellipse gets more cigar-shaped; at exactly $1$ it "breaks open." The two foci of an ellipse are where, if you nailed a string of length $2a$ and pulled it taut with a pencil, the pencil traces the curve — this is the literal gardener's construction.

Worked Examples

Ellipse — Finding Foci and Eccentricity

1. Compare with [6]: $a^2=25\Rightarrow a=5$; $b^2=9\Rightarrow b=3$. Since $a>b$, the major axis is horizontal.
2. Focal distance: $c^2=a^2-b^2=25-9=16\Rightarrow c=4$.
3. Foci lie on the $x$-axis at $(\pm c,0)=(\pm 4,0)$.
4. Eccentricity: $e=c/a=4/5=0.8$.

Parabola — From Focus and Directrix to Equation

1. Vertex is midway between focus and directrix: at $(0,0)$, opening right.
2. Here $p=3$ (focus at $(p,0)$). Use [7]: $y^2=4px$.
3. Substitute: $y^2=4(3)x=12x$.
4. Latus rectum (focal chord perpendicular to axis) has length $|4p|=12$.

Common Misconceptions

A "Circle" That Isn't One

Consider $x^2+y^2-6x+10y+34=0$. It looks like a circle ($A=C$, no $xy$ term). Completing the square: $(x-3{)}^2+(y+5{)}^2=9+25-34=0$. The "radius squared" is $0$ — this is a single point $(3,-5)$, a degenerate conic, not a true circle. Had the constant given a negative right-hand side, there would be no real points at all. The discriminant test [10] classifies the quadratic type but cannot detect such degeneracies.

Connections

Conics tie together vast regions of mathematics. The standard equations are special cases of Quadratic Equations in Two Variables, and recognizing them requires Completing the Square. Their unified projective treatment leads to Projective Geometry, where all non-degenerate conics become a single object. Through the focus–directrix property and Polar Coordinates, every conic gets the compact form $r=\frac{\ell }{1+e\cos \theta }$. In physics, conics are the exact solution curves of the Two-Body Problem under inverse-square gravity, connecting directly to Kepler's Laws of Planetary Motion.

Real-World Applications

1. Planetary & cometary orbits. Kepler's First Law: planets trace ellipses with the Sun at one focus; unbound comets follow parabolic or hyperbolic paths ($e\ge 1$).
2. Satellite dishes & radio telescopes. A parabolic reflector focuses all incoming parallel rays to its focus — the basis of the Arecibo and Green Bank designs.
3. Car headlights & flashlights. A bulb at the focus of a parabolic mirror sends out a parallel beam (the reverse process).
4. Whispering galleries. In elliptical rooms (e.g., St Paul's Cathedral, the U.S. Capitol's Statuary Hall), sound from one focus reflects to the other focus, audible across the room.
5. LORAN & GPS-era hyperbolic navigation. Time-difference of signals from two stations places a ship on a hyperbola (constant distance difference); intersecting hyperbolas fix the position.
6. Cooling towers & arches. Hyperbolic shells (cooling towers) and parabolic/catenary arches exploit conic geometry for structural strength.
7. Lithotripsy. An ellipsoidal reflector concentrates shock waves generated at one focus onto a kidney stone placed at the other focus.

Historical Development

• c. 350 BCE — Menaechmus discovers conics while attacking the problem of doubling the cube; he uses parabola–hyperbola intersections.
• c. 300 BCE — Euclid writes a (now-lost) treatise on conics.
• c. 225 BCE — Apollonius of Perga completes Conics, naming the curves and unifying them from a single cone.
• c. 150 CE — Ptolemy applies conic and related geometry to astronomy in the Almagest.
• 1609 — Johannes Kepler announces that Mars (and all planets) move in ellipses with the Sun at a focus (Kepler's First Law).
• 1637 — René Descartes and Pierre de Fermat introduce coordinate geometry, recasting conics as second-degree equations.
• 1687 — Isaac Newton proves in the Principia that inverse-square gravity forces conic-section orbits, deriving Kepler's laws.
• 1822 — Germinal Pierre Dandelin introduces the elegant Dandelin spheres, proving the cone-slice and focus definitions coincide.
• 19th c. — Poncelet, Steiner, and others develop the projective theory, in which the five conic types become one.

Subtleties & Limits

While the classification of real conics is fully settled, subtleties remain pedagogically and mathematically rich. Degenerate conics (points, single/double lines, empty sets) all satisfy [9] but escape the locus definitions — a unified treatment requires projective geometry, where conics over $\mathbb{C}$ are classified more cleanly than over $\mathbb{R}$. The general conic-fitting problem (finding the best-fit ellipse to noisy data points) is numerically delicate: ordinary least squares can return hyperbolas instead of ellipses, prompting constrained algorithms (Fitzgibbon's direct ellipse fit, 1999) still actively refined. In higher dimensions, the analogous quadric surfaces (ellipsoids, paraboloids, hyperboloids) have a richer classification whose interaction with optimization and computer vision remains an area of ongoing engineering research.

See Also