1. Slicing a double cone
Take two cones joined apex-to-apex, opening up and down. Pass a flat plane through this double cone. The intersection — the curve traced on the plane — is one of four shapes, and which one you get depends entirely on the angle of the cut.
- Plane perpendicular to the cone's axis: circle.
- Plane tilted, but still cutting only one nappe entirely: ellipse.
- Plane parallel to a slant edge (a generator) of the cone: parabola.
- Plane steep enough to cut both nappes: hyperbola (two disconnected branches).
That's it — one cone, one plane, four families. The pictures below show all four cuts on the same cone.
Same cone, four cuts. The angle between the slicing plane and the cone's axis decides which curve appears.
If the plane passes through the cone's apex you get a degenerate conic instead of a curve: a single point (just the apex), a line (one generator), or a pair of intersecting lines (both generators of a steep cut). These are the boundary cases between the four main families.
2. The unified definition: focus, directrix, eccentricity
The cone story is beautiful, but it's not how you usually work with conics. There's a second definition — purely in the plane, no cone in sight — that turns out to be equivalent and far more useful for computation.
Fix a point $F$ (the focus) and a line $\ell$ (the directrix) not passing through $F$. Fix a positive constant $e$ (the eccentricity). The conic is the set of points $P$ for which
$$ \frac{\text{distance from } P \text{ to } F}{\text{distance from } P \text{ to } \ell} = e. $$Different values of $e$ trace out different families:
- $e = 0$ — circle (degenerate case of the ratio; the focus is the centre)
- $0 < e < 1$ — ellipse
- $e = 1$ — parabola
- $e > 1$ — hyperbola
One parameter, one definition, four families. As $e$ slides from $0$ upward, the curve morphs continuously: a circle stretches into an ellipse, the ellipse elongates until it bursts open into a parabola, and beyond $e = 1$ the curve splits into the two branches of a hyperbola. The same continuous deformation underlies the polar equation
$$ r = \frac{e p}{1 - e \cos\theta} $$with the focus at the origin. This single formula generates every non-circular conic — and it's how astronomers naturally describe orbits.
Eccentricity is "how un-circular" the curve is. Zero is perfectly round. Approaching $1$ from below, an ellipse stretches into a cigar. At exactly $1$, the far end of the cigar opens up and you have a parabola. Past $1$, the curve has flown apart into two branches that retreat toward straight-line asymptotes.
3. Circle
The simplest conic. A circle is the set of points at fixed distance $r$ (the radius) from a fixed centre $(h, k)$.
$$ (x - h)^2 + (y - k)^2 = r^2 $$The metric definition uses just one focus (the centre) and zero distance ratio — eccentricity $e = 0$. Every other conic has the geometric "tension" of being pulled between a focus and a directrix; the circle has none, which is exactly why it's symmetric in every direction.
Circle
Equation: $(x-h)^2 + (y-k)^2 = r^2$. Eccentricity: $e = 0$.
Worth knowing: a circle is the special ellipse where the two foci have collapsed onto each other — both at the centre. The major and minor "axes" become equal, and the constant focal-sum definition reduces to "constant distance from one point."
4. Ellipse
The set of points whose distances to two fixed foci sum to a fixed constant $2a$.
If you pin two thumbtacks to a board, loop a string around them, and trace with a pencil that keeps the string taut, you draw an ellipse. The thumbtacks are the foci; the loop length encodes $2a$.
Centred at the origin with horizontal major axis, the standard equation is
$$ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1, \qquad a > b > 0. $$The numbers $a$ and $b$ are the semi-major and semi-minor axes — half the lengths of the two principal diameters. The foci sit on the major axis at distance $c$ from centre, where
$$ c^2 = a^2 - b^2, \qquad e = \frac{c}{a} \in (0, 1). $$As $b \to a$, $c \to 0$ and the ellipse rounds out into a circle. As $b \to 0$, $c \to a$, eccentricity climbs toward $1$, and the ellipse flattens into a sliver along the major axis.
For any point $P$ on an ellipse, the two distances to the foci add up to the same value $2a$.
Numerical example: $\dfrac{x^2}{25} + \dfrac{y^2}{9} = 1$ has $a = 5$, $b = 3$, so $c = \sqrt{25 - 9} = 4$. Foci $(\pm 4, 0)$, eccentricity $e = 4/5 = 0.8$ — quite elongated.
5. Parabola
The set of points equidistant from a fixed point (the focus) and a fixed line (the directrix). Eccentricity $e = 1$.
The parabola is the borderline case between ellipse and hyperbola. The "second focus" of an ellipse has effectively wandered off to infinity, leaving a single focus and a single directrix that play symmetric roles.
Standard form, vertex at the origin and opening upward:
$$ x^2 = 4p\, y. $$Here $p$ is the focal parameter — the directed distance from vertex to focus. The focus is at $(0, p)$ and the directrix is the line $y = -p$. Open downward by flipping the sign of $p$; open sideways by swapping $x$ and $y$.
A parabola is the locus of points equidistant from focus $F$ and directrix line.
Worked: $y^2 = 8x$. Compare with $y^2 = 4px$ to read $4p = 8$, so $p = 2$. Vertex at the origin, opening to the right; focus $(2, 0)$, directrix $x = -2$.
6. Hyperbola
The set of points whose distances to two fixed foci have a constant absolute difference $2a$. Eccentricity $e > 1$.
An ellipse uses the sum of focal distances; a hyperbola uses the difference. That one sign flip is why the hyperbola breaks open into two disconnected branches — one branch is closer to each focus.
Centred at the origin with horizontal transverse axis:
$$ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1. $$The vertices are at $(\pm a, 0)$ — these are the points where each branch is closest to the centre. The foci sit further out at $(\pm c, 0)$, where
$$ c^2 = a^2 + b^2, \qquad e = \frac{c}{a} > 1. $$Note the plus sign — opposite to the ellipse. The number $b$ doesn't show up as a vertex on the $y$-axis (the curve never crosses there), but it controls something equally important: the asymptotes, the two straight lines the branches approach at infinity:
$$ y = \pm \frac{b}{a}\, x. $$Far from the centre, the hyperbola is almost indistinguishable from these two lines; up close, it bows in toward the vertices.
A hyperbola has two branches, two foci, and two asymptotes.
Worked: $\dfrac{x^2}{16} - \dfrac{y^2}{9} = 1$. Here $a = 4$, $b = 3$, $c = \sqrt{16 + 9} = 5$. Vertices $(\pm 4, 0)$, foci $(\pm 5, 0)$, asymptotes $y = \pm \tfrac{3}{4} x$, eccentricity $5/4$.
One curiosity: when $a = b$, the asymptotes are perpendicular and the curve is called a rectangular hyperbola. Rotating by $45°$ turns its equation into the familiar $xy = c$ — the shape of $y = 1/x$.
7. Classifying a general quadratic
Hand someone a circle equation and they'll spot it instantly. But what about something messier, like $4x^2 + 9y^2 + 24x - 36y + 36 = 0$? Or $x^2 - 4xy + y^2 = 5$?
Every conic — no matter how rotated, shifted, or scaled — fits the general second-degree equation
$$ A x^2 + B x y + C y^2 + D x + E y + F = 0. $$The good news: you don't have to manipulate it into standard form to know which family it belongs to. Compute the discriminant:
$$ \Delta = B^2 - 4AC. $$The sign of $\Delta$ tells you the type, no matter how the equation is rotated:
| Discriminant | Conic type | Picture |
|---|---|---|
| $B^2 - 4AC < 0$ | Ellipse (circle if also $A = C$, $B = 0$) | Closed loop |
| $B^2 - 4AC = 0$ | Parabola | Single open curve |
| $B^2 - 4AC > 0$ | Hyperbola | Two branches |
Quick check: for $x^2 + y^2 - 4 = 0$ we have $A = 1$, $B = 0$, $C = 1$, so $\Delta = -4 < 0$ — ellipse (a circle, in fact). For $y^2 - 8x = 0$, $A = 0$, $C = 1$, $\Delta = 0$ — parabola.
When the $xy$ term is present, the conic is rotated relative to the coordinate axes; you can derotate by an angle $\theta$ with $\cot(2\theta) = (A - C)/B$ to land in a frame where the standard equations apply. But you don't have to derotate just to classify — the discriminant is invariant under rotation.
Where does the discriminant come from?
The quadratic part $A x^2 + B x y + C y^2$ is a quadratic form. Diagonalising it (rotating to the eigenvector frame) gives $\lambda_1 X^2 + \lambda_2 Y^2$, where $\lambda_1$ and $\lambda_2$ are the eigenvalues of the matrix $\begin{pmatrix} A & B/2 \\ B/2 & C \end{pmatrix}$. The determinant is $AC - B^2/4 = -(B^2 - 4AC)/4$, and its sign tells you whether the two eigenvalues agree in sign (ellipse), disagree (hyperbola), or include a zero (parabola). That's the discriminant up to a constant.
8. Why the world is full of conics
Kepler: orbits are ellipses
The most famous appearance. Kepler's first law (1609) states that every planet orbits the Sun on an ellipse with the Sun at one focus — not at the centre. Newton later proved this from his inverse-square law of gravitation. Bound orbits are ellipses; barely-escaping trajectories are parabolas; comets that swing past once and leave for ever are on hyperbolic paths. All three are the same kind of curve, separated only by eccentricity.
Projectiles trace parabolas
Throw a stone. Ignore air resistance. Constant downward gravity plus constant horizontal velocity produces a parabolic trajectory — directly from $y = -\tfrac{1}{2} g t^2 + v_y t$ paired with $x = v_x t$. Galileo worked this out in the early 17th century. Every artillery shell, every basketball, every dropped coin traces a parabola.
Reflectors and whispering galleries
Each conic has a beautiful reflective property:
- Parabola. Rays parallel to the axis reflect off the curve and converge on the focus. This is why satellite dishes, headlight reflectors, and radio-telescope mirrors are paraboloidal — they concentrate parallel incoming signals onto a single receiver.
- Ellipse. Rays from one focus reflect off the curve and converge on the other focus. Whispering galleries exploit this — speak softly at one focus, and a listener at the other focus hears you clearly across a noisy room. Medical lithotripsy uses elliptical reflectors to focus shock waves onto kidney stones.
- Hyperbola. A ray aimed at one focus reflects off the convex side toward the other focus. The Cassegrain telescope design uses a hyperbolic secondary mirror to redirect light gathered by a parabolic primary.
Shadows, dishes, sundials
Hold a flashlight (a cone of light) against a wall and the bright edge traces a hyperbola — you've physically performed a conic section. The shadow boundary on the wall of a conical lampshade is the same. Sundial hour curves are conics in the right setups.
If parabolas focus parallel light, they also do the reverse: light placed at the focus reflects out in a parallel beam. That's the entire optical design of a flashlight or car headlight.
9. Common pitfalls
If the larger denominator is under $x^2$, the major axis is horizontal; if under $y^2$, vertical. The foci always lie on the major axis. Don't memorise "$a$ is the bigger one" without also locating it — the layout of the foci depends on it.
For an ellipse $c^2 = a^2 - b^2$; for a hyperbola $c^2 = a^2 + b^2$. The sign is opposite. The foci of a hyperbola sit farther from centre than the vertices ($c > a$), reflecting $e > 1$.
$(x - h)^2 = 4p(y - k)$ opens upward if $p > 0$ and downward if $p < 0$. $(y - k)^2 = 4p(x - h)$ opens right if $p > 0$, left if $p < 0$. Don't confuse the squared variable with the direction — it's the linear one that points the way.
$x^2 + y^2 = -1$ has no real points at all. The locus is genuinely empty — it isn't "the imaginary circle." Similar things happen when completing the square gives a negative constant where you needed a positive one. Always sanity-check that the equation actually has solutions before naming the curve.
$e \geq 0$ for every conic; it's a ratio of two distances. A sign error in $c$ shouldn't propagate to $e$ — take the absolute value.
10. Worked examples
Try each before opening the solution. The goal is to recognise the shape from the equation, then read off the key features.
Example 1 · Classify $\dfrac{x^2}{9} + \dfrac{y^2}{4} = 1$ and find its foci.
Shape. Both squared terms have the same sign and different denominators — an ellipse. The larger denominator is under $x^2$, so the major axis is horizontal.
Read parameters. $a^2 = 9$, $b^2 = 4$, so $a = 3$, $b = 2$.
Foci. $c^2 = a^2 - b^2 = 9 - 4 = 5$, so $c = \sqrt{5}$. Foci at $(\pm\sqrt{5}, 0)$. Eccentricity $e = \sqrt{5}/3 \approx 0.745$.
Example 2 · Find the focus and directrix of $y^2 = 8x$.
Shape. One variable squared, the other linear — parabola. The $y$ is squared, so the parabola opens along the $x$-axis.
Match to standard form. Compare $y^2 = 8x$ with $y^2 = 4px$. Then $4p = 8$, so $p = 2$.
Focus, directrix. Focus at $(p, 0) = (2, 0)$; directrix $x = -p = -2$. Vertex at origin; opens to the right.
Example 3 · For $\dfrac{x^2}{16} - \dfrac{y^2}{9} = 1$, find vertices, foci, and asymptotes.
Shape. Squared terms with opposite signs — hyperbola, opening left/right.
Parameters. $a^2 = 16$, $b^2 = 9$, so $a = 4$, $b = 3$. $c^2 = a^2 + b^2 = 25$, so $c = 5$.
Vertices. $(\pm 4, 0)$. Foci. $(\pm 5, 0)$. Asymptotes. $y = \pm \tfrac{b}{a}x = \pm \tfrac{3}{4}x$. Eccentricity. $e = 5/4 = 1.25$.
Example 4 · Classify $4x^2 + 4xy + y^2 - 6x + 5 = 0$ using the discriminant.
Read off coefficients. $A = 4$, $B = 4$, $C = 1$.
Discriminant. $B^2 - 4AC = 16 - 16 = 0$.
Conclusion. A parabola, rotated relative to the axes (because $B \neq 0$). Standard form requires rotating by $\theta$ with $\cot(2\theta) = (A - C)/B = 3/4$, but classification is done.
Example 5 · Build the ellipse with foci $(\pm 3, 0)$ and sum of focal distances $10$.
Read off $a$ and $c$. $2a = 10$ gives $a = 5$. The foci are at $(\pm c, 0)$, so $c = 3$.
Compute $b$. $b^2 = a^2 - c^2 = 25 - 9 = 16$, so $b = 4$.
Equation. $\dfrac{x^2}{25} + \dfrac{y^2}{16} = 1$. Eccentricity $e = 3/5 = 0.6$.