Conic Sections

Take a double cone — two ice-cream cones joined tip to tip — and slice it with a flat plane. Depending on the angle of the cut, the edge of the slice traces one of four curves: a circle, an ellipse, a parabola, or a hyperbola. That single geometric idea unites shapes that at first seem unrelated: the orbit of a planet, the path of a thrown ball, the reflector behind a flashlight bulb, and the cross-section of a satellite dish. These are the conic sections, and they are among the most useful curves in all of mathematics.

What makes conics beautiful is that the same family can be described three completely different ways — as slices of a cone, as equations in $x$ and $y$, and as sets of points obeying a distance rule involving special points called foci. Understanding how those three descriptions line up is the whole game, and it turns "memorizing four formulas" into "seeing one idea from four angles."

Learning Objectives

• Identify each conic section as a specific slice of a double cone.
• Recognize and manipulate the standard equations of the circle, ellipse, parabola, and hyperbola.
• Define and locate the focus, directrix, and center of each conic.
• Understand eccentricity as the single number that classifies every conic.
• Solve worked problems involving foci, vertices, and axes.
• Connect conics to real orbits, optics, and engineering.

Quick Answer

A conic section is any curve formed by intersecting a plane with a double cone. If the plane is perpendicular to the cone's axis you get a circle; tilt it slightly and you get an ellipse; tilt until it is parallel to the cone's side and you get a parabola; tilt further so it cuts both halves of the cone and you get a hyperbola. Every conic can also be defined by its eccentricity $e$, the ratio of distance-to-focus over distance-to-directrix: a circle has $e = 0$, an ellipse has $0 < e < 1$, a parabola has $e = 1$, and a hyperbola has $e > 1$. In coordinates, all four are special cases of the general second-degree equation $Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0$.

Where It Came From

Conic sections were first studied seriously by Greek geometers around 350 BCE. Menaechmus is credited with discovering them while attacking the classic problem of doubling the cube — constructing a cube with twice the volume of a given one. That problem reduces to finding certain mean proportionals, which Menaechmus realized could be located as the intersection points of curves we now call the parabola and hyperbola. So conics were born not as abstract art but as a tool to solve a stubborn construction problem straightedge and compass could not.

The subject reached its classical peak with Apollonius of Perga (c. 262–190 BCE), whose eight-book treatise Conics was so complete it remained the definitive work for nearly two thousand years. Apollonius gave us the very names ellipse, parabola, and hyperbola, choosing them from Greek words meaning "falling short," "applying exactly," and "exceeding" — describing how each curve's defining area compared to a reference rectangle. He studied conics purely as geometry, with no equations and no coordinates.

For centuries conics were admired but seemed to have little practical use. That changed dramatically in 1609, when Johannes Kepler, analyzing Tycho Brahe's meticulous observations of Mars, announced that planets do not move in circles but in ellipses, with the Sun at one focus. This was revolutionary: it broke two thousand years of belief in perfect circular motion and demanded the ellipse as the true shape of the heavens. Around the same time, Galileo showed that a projectile under gravity follows a parabola, and it was later understood that a parabolic mirror focuses all incoming parallel light to a single point — the principle behind reflecting telescopes, headlights, and satellite dishes. What began as Greek geometry became the language of astronomy and optics. Descartes and Fermat, developing coordinate geometry in the 1630s, finally connected the curves to algebraic equations, giving us the standard forms we use today.

Slices of a Cone: The Unifying Picture

Imagine a double cone standing on its point, with a vertical axis, and a plane cutting through it. The angle between the cutting plane and the cone's axis determines everything.

• Circle — plane perpendicular to the axis (horizontal cut).
• Ellipse — plane tilted, but still cutting only one nappe (one half of the cone) completely across.
• Parabola — plane tilted exactly parallel to a line on the cone's surface (a "slant edge").
• Hyperbola — plane steep enough to slice through both nappes, giving two separate branches.

If the plane passes exactly through the vertex, you get degenerate conics: a single point, a single line, or a pair of crossing lines. This one mental image explains why the four curves belong to a single family: they differ only by the tilt of a knife.

The Circle and the Ellipse

A circle is the set of all points at a fixed distance $r$ (the radius) from a center $(h, k)$. Its equation is:

$(x - h)^2 + (y - k)^2 = r^2$

An ellipse is a "stretched circle." Formally it is the set of points for which the sum of distances to two fixed points (the foci) is constant. If you pin a loop of string around two tacks and trace it taut with a pencil, you draw an ellipse — the tacks are the foci. Its standard equation, centered at the origin with horizontal major axis, is:

$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1, \qquad a > b > 0$

Here $a$ is the semi-major axis (half the long width) and $b$ is the semi-minor axis. The foci sit on the major axis at $(\pm c, 0)$, where $c^2 = a^2 - b^2$. The eccentricity is $e = c/a$, always between 0 and 1; as $e \to 0$ the ellipse rounds into a circle, and as $e \to 1$ it stretches long and thin.

Worked example. Find the foci and eccentricity of $\dfrac{x^2}{25} + \dfrac{y^2}{9} = 1$.

Here $a^2 = 25$ so $a = 5$, and $b^2 = 9$ so $b = 3$. Then $c^2 = a^2 - b^2 = 25 - 9 = 16 \implies c = 4.$ The foci are at $(\pm 4, 0)$, and the eccentricity is $e = c/a = 4/5 = 0.8$. Since $0.8 < 1$, this is a genuinely elongated ellipse. As a check on the defining property: the vertex $(5, 0)$ has distances $5 - 4 = 1$ and $5 + 4 = 9$ to the two foci, summing to $10 = 2a$ — exactly as required.

The Parabola

A parabola is the set of all points equidistant from a fixed point (the focus) and a fixed line (the directrix). This single balance condition gives the curve its remarkable focusing property. The standard equation of a parabola opening upward with vertex at the origin is:

$x^2 = 4py$

where $p$ is the distance from the vertex to the focus. The focus sits at $(0, p)$ and the directrix is the line $y = -p$. A parabola has eccentricity exactly $e = 1$: distance-to-focus equals distance-to-directrix for every point on the curve.

Worked example. For the parabola $x^2 = 12y$, find the focus and directrix, and verify the point $(6, 3)$ lies on it.

Matching $x^2 = 4py$ with $x^2 = 12y$ gives $4p = 12$, so $p = 3$. The focus is at $(0, 3)$ and the directrix is $y = -3$. Check the point $(6, 3)$: distance to the focus is $\sqrt{(6-0)^2 + (3-3)^2} = \sqrt{36} = 6,$ and distance to the directrix $y = -3$ is $3 - (-3) = 6$. The two distances are equal, confirming $(6,3)$ is on the parabola.

The reflective property follows from this geometry: any ray traveling parallel to the axis reflects off the parabola straight through the focus. That is exactly why a satellite dish gathers weak parallel signals to a receiver at the focus, and — run in reverse — why a headlight with a bulb at the focus throws a parallel beam.

The Hyperbola

A hyperbola is the set of points for which the difference of distances to two foci is constant. Where the ellipse uses a sum, the hyperbola uses a difference — and that flip produces two separate opening branches instead of one closed loop. The standard equation, opening left–right, is:

$\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$

The vertices are at $(\pm a, 0)$, the foci at $(\pm c, 0)$ where now $c^2 = a^2 + b^2$ (note the plus sign — the opposite of the ellipse). Its eccentricity $e = c/a$ is always greater than 1. Far from the center, the curve hugs two straight asymptotes, the lines $y = \pm \dfrac{b}{a} x$.

Worked example. For $\dfrac{x^2}{9} - \dfrac{y^2}{16} = 1$, find the vertices, foci, asymptotes, and eccentricity.

Here $a^2 = 9$ so $a = 3$, and $b^2 = 16$ so $b = 4$. Then $c^2 = a^2 + b^2 = 9 + 16 = 25 \implies c = 5.$ Vertices: $(\pm 3, 0)$. Foci: $(\pm 5, 0)$. Asymptotes: $y = \pm \tfrac{4}{3} x$. Eccentricity: $e = c/a = 5/3 \approx 1.67$, which is greater than 1 as expected for a hyperbola.

Eccentricity: One Number to Rule Them All

The deepest unifying idea is the focus–directrix definition: fix a focus point and a directrix line, and consider all points $P$ where the ratio $e = \frac{\text{distance from } P \text{ to focus}}{\text{distance from } P \text{ to directrix}}$ is constant. That constant $e$ is the eccentricity, and its value alone decides the shape.

Eccentricity $e$	Conic	Cone slice
$e = 0$	Circle	Perpendicular to axis
$0 < e < 1$	Ellipse	Tilted, one nappe
$e = 1$	Parabola	Parallel to cone edge
$e > 1$	Hyperbola	Cuts both nappes

This is why conics form one family and not four separate curiosities: turn a single dial from 0 upward and you pass smoothly from circle to ellipse to parabola to hyperbola.

Real-World Applications

• Planetary and satellite orbits. Planets, moons, and satellites trace ellipses with the central body at one focus (Kepler's First Law). Comets on unbound trajectories follow parabolas or hyperbolas; a spacecraft doing a slingshot flyby travels a hyperbolic path relative to the planet.
• Optics and communications. Parabolic mirrors focus parallel rays to a point — reflecting telescopes, solar concentrators, satellite dishes, and microphones. Run in reverse, they produce parallel beams in searchlights and car headlights.
• Architecture and engineering. A hanging chain forms a catenary (close cousin of a parabola), and arches and suspension-bridge cables use parabolic profiles for even load distribution. Whispering galleries use elliptical ceilings so a whisper at one focus is heard clearly at the other.
• Navigation and location. LORAN and modern time-difference-of-arrival systems use the constant-difference property of hyperbolas to pinpoint a receiver's position from signal timing.
• Medicine. A lithotripter uses an ellipsoidal reflector to focus shock waves generated at one focus onto a kidney stone placed at the other, shattering it without surgery.

Common Mistakes

Mistake 1: Using $c^2 = a^2 - b^2$ for a hyperbola. Students memorize the ellipse relation and reuse it. For an ellipse $c^2 = a^2 - b^2$, but for a hyperbola $c^2 = a^2 + b^2$. The sign flips because the hyperbola's foci lie beyond its vertices, so $c > a$ always. Remember: ellipse subtracts, hyperbola adds.

Mistake 2: Thinking $a$ is always the $x$-denominator. In $\frac{x^2}{9} + \frac{y^2}{25} = 1$, the larger denominator is under $y^2$, so the major axis is vertical and $a^2 = 25$, $b^2 = 9$. Always let $a$ be the larger of the two, then the major axis lies along the variable that carries $a^2$. Blindly reading $a^2$ off the $x$-term gives the wrong foci.

Mistake 3: Confusing the parabola's $p$ with a squared quantity. In $x^2 = 4py$, the coefficient is $4p$, not $p$ or $p^2$. Given $x^2 = 12y$, some write $p = 12$ or $p = \sqrt{12}$. Correctly, $4p = 12 \Rightarrow p = 3$. The focus is at distance $p$, so getting this factor of 4 wrong misplaces the focus entirely.

Comparison and Connections

The four conics are best seen side by side. The circle is really an ellipse with $a = b$ (both foci merged at the center), and the parabola is the knife-edge case sitting exactly between the bounded ellipse and the unbounded hyperbola.

Feature	Circle	Ellipse	Parabola	Hyperbola
Eccentricity	$0$	$0<e<1$	$1$	$e>1$
Foci	1 (center)	2	1	2
Distance rule	fixed distance	sum constant	focus = directrix	difference constant
$c$ relation	—	$c^2=a^2-b^2$	—	$c^2=a^2+b^2$
Branches	1 closed	1 closed	1 open	2 open
Asymptotes	none	none	none	two

All four are unified by the general second-degree equation $Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0$. The discriminant $B^2 - 4AC$ classifies it: negative gives an ellipse (or circle), zero a parabola, and positive a hyperbola — the algebraic echo of the eccentricity test.

Practice Questions

Recall

Q. What curve results from slicing a double cone with a plane perpendicular to its axis, and what is its eccentricity? A. A circle, with eccentricity $e = 0$.

Understanding

Q. Explain why a hyperbola has $c > a$ while an ellipse has $c < a$. A. For an ellipse the foci lie inside the curve between the vertices, so the focal distance $c$ is less than the semi-major axis $a$ (and $c^2 = a^2 - b^2$). For a hyperbola the foci lie outside the vertices, so $c > a$ (and $c^2 = a^2 + b^2$). The reversed relationship reflects whether the defining rule uses a sum (bounded) or a difference (unbounded).

Application

Q. Find the foci and eccentricity of the ellipse $\dfrac{x^2}{16} + \dfrac{y^2}{7} = 1$. A. $a^2 = 16, b^2 = 7$, so $c^2 = 16 - 7 = 9$, giving $c = 3$. Foci at $(\pm 3, 0)$; eccentricity $e = 3/4 = 0.75$.

Analysis

Q. A parabolic satellite dish has equation $x^2 = 20y$ (units in feet). Where should the receiver be placed, and why there? A. Match $4p = 20 \Rightarrow p = 5$, so the focus is at $(0, 5)$ — five feet above the vertex. The receiver goes at the focus because the parabola's reflective property sends every incoming ray parallel to the axis straight to the focus, concentrating the faint signal at that single point.

FAQ

Why are all four curves called by one name if they look so different? Because they are literally slices of the same object — a double cone — differing only by the tilt of the cutting plane, and because a single number, eccentricity, moves continuously through all four. They are four faces of one idea.

Is a circle really just a special ellipse? Yes. When the two foci of an ellipse coincide at the center, $a = b$, $c = 0$, and $e = 0$ — the equation becomes $x^2 + y^2 = r^2$. Every circle is an ellipse of zero eccentricity.

What is the directrix, and does every conic have one? The directrix is a fixed line used in the focus–directrix definition; a point is on the conic when its distance to the focus is $e$ times its distance to the directrix. Parabolas, ellipses, and hyperbolas all have directrices. A true circle is the limiting case where the directrix recedes to infinity.

How did Kepler know orbits were ellipses and not circles? Tycho Brahe's observations of Mars were accurate to about two arcminutes. Kepler's best circular model missed by about eight arcminutes — small, but larger than Brahe's known error. Rather than dismiss the gap, Kepler trusted the data, and only an ellipse with the Sun at a focus fit it. It was a triumph of taking small discrepancies seriously.

Why does a parabolic mirror focus light to a point but a spherical one does not? The parabola is defined by the equidistant focus–directrix property, which guarantees that every ray parallel to the axis reflects exactly through the focus. A sphere only approximates this near its center; farther out, parallel rays miss the focus — an error called spherical aberration, which is why precision telescopes use parabolic mirrors.

Quick Revision

• Conics = plane slicing a double cone: circle, ellipse, parabola, hyperbola.
• Eccentricity classifies all: $e=0$ circle, $0<e<1$ ellipse, $e=1$ parabola, $e>1$ hyperbola.
• Circle: $(x-h)^2 + (y-k)^2 = r^2$.
• Ellipse: $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$, foci $(\pm c,0)$, $c^2=a^2-b^2$, $e=c/a$.
• Parabola: $x^2=4py$, focus $(0,p)$, directrix $y=-p$, $e=1$.
• Hyperbola: $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$, foci $(\pm c,0)$, $c^2=a^2+b^2$, asymptotes $y=\pm\frac{b}{a}x$.
• Ellipse = sum of focal distances constant; hyperbola = difference constant; parabola = focus-directrix equal.
• Discriminant $B^2-4AC$: $<0$ ellipse, $=0$ parabola, $>0$ hyperbola.

Related Topics

Prerequisites

• Geometry overview

Related Topics

• Coordinate geometry and the distance formula
• Quadratic equations and second-degree curves

Next Topics

• Polar coordinates and the polar form of conics
• Orbital mechanics and Kepler's laws