Conic Sections — Parabola, Ellipse, Hyperbola, Circle

⚡ Key Takeaways

Conic sections are the four curves — circle, ellipse, parabola, hyperbola — obtained by slicing a double cone with a plane. Each is the locus of points whose distances to fixed points (foci) or a point and line (focus/directrix) satisfy a fixed relationship. They govern planetary orbits, projectile paths, reflector shapes, and satellite dishes. Studied since Menaechmus (~350 BCE) and named by Apollonius of Perga (~200 BCE).

💡 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),

1

\quad \frac{\text{dist}(P,F)}{\text{dist}(P,D)} = e

Derivation hint: This single ratio condition, with $e<1$ , $e=1$ , $e>1$ , generates ellipse, parabola, hyperbola respectively; the circle is the limit $e\to 0$ . The geometry of the cone and the algebra of the quadratic equation are two faces of this one locus condition.

DefinitionThe 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 $2a$ :

2

\quad |PF_1| + |PF_2| = 2a, \qquad 2a > |F_1F_2|

Derivation hint: Take the constant string-length condition; $a$ is the semi-major axis.

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

3

\quad |PF| = \text{dist}(P,D)

Derivation hint: Set $e=1$ in equation [1].

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

4

\quad \big|\,|PF_1| - |PF_2|\,\big| = 2a, \qquad 0 < 2a < |F_1F_2|

Derivation hint: The two branches arise from the two signs of the difference.

DefinitionStandard-Form Equations

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

Circle (center origin, radius $r$ ):

5

\quad x^2 + y^2 = r^2

Derivation hint: Apply the distance formula to $|PC|=r$ with $C=(0,0)$ .

Ellipse (center origin, $a>b>0$ ):

6

\quad \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1, \qquad c^2 = a^2 - b^2

Derivation hint: Place foci at $(\pm c,0)$ , expand $|PF_1|+|PF_2|=2a$ , and simplify with $b^2=a^2-c^2$ .

Parabola (vertex origin, opening right):

7

\quad y^2 = 4px

Derivation hint: Put focus at $(p,0)$ , directrix $x=-p$ , and square $|PF|=\text{dist}(P,D)$ .

Hyperbola (center origin, opening left/right):

8

\quad \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1, \qquad c^2 = a^2 + b^2

Derivation hint: Place foci at $(\pm c,0)$ , expand $\big||PF_1|-|PF_2|\big|=2a$ , set $b^2=c^2-a^2$ . Here $a$ = semi-major (or transverse) axis, $b$ = semi-minor (or conjugate) axis, $c$ = focal distance from center, $p$ = focal parameter, $r$ = radius.

TheoremGeneral Second-Degree Equation & the Discriminant

Every conic is described by

9

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

Derivation hint: Expand any standard form after rotation and translation; cross-term $Bxy$ encodes rotation of the axes. The type is classified by the discriminant $\Delta = B^2 - 4AC$ :

10

\quad \Delta = B^2 - 4AC \begin{cases} <0 & \text{ellipse (circle if } A=C,\ B=0) \\ =0 & \text{parabola} \\ >0 & \text{hyperbola} \end{cases}

Derivation hint: $\Delta$ is invariant under rotation; its sign matches the eigenvalue signs of the quadratic-form matrix $\begin{pmatrix}A & B/2\\ B/2 & C\end{pmatrix}$ .

ProofDeriving 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: $(x+c)^2+y^2 = 4a^2 - 4a\sqrt{(x-c)^2+y^2} + (x-c)^2+y^2$ . 5. Expand and cancel: $4cx = 4a^2 - 4a\sqrt{(x-c)^2+y^2}$ , so $\sqrt{(x-c)^2+y^2} = a - \frac{c}{a}x$ . 6. Square again: $(x-c)^2 + y^2 = a^2 - 2cx + \frac{c^2}{a^2}x^2$ . 7. Expand left side and simplify: $x^2\left(1-\frac{c^2}{a^2}\right) + y^2 = a^2 - c^2$ . 8. Let $b^2 = a^2 - c^2$ . Then $x^2\frac{b^2}{a^2} + y^2 = b^2$ . 9. Divide by $b^2$ : $\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1.$ $\blacksquare$

∎

ProofEccentricity 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 $\sqrt{(x-c)^2+y^2} = e\left|x - \tfrac{c}{e^2}\right|$ . 2. Square: $(x-c)^2 + y^2 = e^2 x^2 - 2cx + \tfrac{c^2}{e^2}$ . 3. Collect: $(1-e^2)x^2 + y^2 = \tfrac{c^2}{e^2}(1-e^2)$ (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. $\blacksquare$

∎

Figure

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

mathematician · 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.

🧠 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.

ExampleEllipse — Finding Foci and Eccentricity

Given: $\dfrac{x^2}{25} + \dfrac{y^2}{9} = 1$ .

Find: semi-axes, foci, eccentricity.

Steps: 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: $e = c/a = 4/5 = 0.8$ .

Answer:

\boxed{a=5,\ b=3,\ \text{foci }(\pm 4,0),\ e=0.8}

(All lengths in the same units as the coordinates.)

ExampleParabola — From Focus and Directrix to Equation

Given: A parabola with focus $F=(3,0)$ and directrix $x = -3$ .

Find: its equation and the length of the latus rectum.

Steps: 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 $|4p| = 12$ .

Answer:

\boxed{y^2 = 12x,\quad \text{latus rectum} = 12 \text{ units}}

⚠️ Common mistake

The semi-minor axis is NOT the focal distance. For an ellipse, $c^2 = a^2 - b^2$ (foci inside), but for a hyperbola $c^2 = a^2 + b^2$ (foci outside the vertices). Mixing these sign conventions is the single most common error. Also: in $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ , the larger denominator always sits under the variable along the major axis — if $b^2>a^2$ the major axis is vertical, so don't blindly call the $x$ -term's denominator $a^2$ .

CounterexampleA "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.

connection

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.

🔬 Where this appears

⏳ History

- 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.

open-problem

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.

Conic Sections — Parabola, Ellipse, Hyperbola, Circle

Formal Statement

Why It Holds / Derivation

Building Intuition

Worked Examples

Common Misconceptions

Connections

Real-World Applications

Historical Development

Subtleties & Limits

See Also