Take a cone — an ice-cream cone, the spreading beam of a flashlight — and slice it with a flat plane. Tilt the plane and the edge of the cut morphs: from a circle, to an ellipse, to a parabola, to a hyperbola. Four famous curves, all carved from one solid. That shared origin is exactly why they share so much algebra.
One cone, four curves
Imagine a double cone — two cones tip to tip, opening in opposite directions (mathematicians call each half a nappe). Now bring in a flat plane and watch what the intersection traces as you change the plane's angle.
- Horizontal slice — cut straight across, perpendicular to the axis, and you get a perfect circle.
- Gentle tilt — tip the plane a little and the circle stretches into an ellipse, an oval pinched along one direction.
- Parallel to the side — tilt until the plane runs exactly parallel to the slope of the cone, and the curve opens up: it can no longer close, and you get a parabola.
- Steeper still — tip the plane past that angle so it cuts through both nappes, and the curve breaks into two opposing branches — a hyperbola.
Nothing about the cone changed. Only the angle of the slice did. The same surface gives you all four curves in turn, which is the first hint that these shapes are far more closely related than they look.
One number tells them apart: eccentricity
There is a second way to define every one of these curves — one that needs no cone at all, just a special point and a special line. Fix a point called the focus and a line called the directrix. Then a conic is the set of all points whose distance to the focus is \( e \) times its distance to the directrix:
\[ \frac{\text{distance to focus}}{\text{distance to directrix}} = e. \]That single ratio \( e \) is the eccentricity, and remarkably it alone names the curve:
- Circle: \( e = 0 \) — perfectly round, no stretch at all.
- Ellipse: \( 0 < e < 1 \) — closed but flattened.
- Parabola: \( e = 1 \) — focus and directrix held in perfect balance.
- Hyperbola: \( e > 1 \) — open, with two branches.
Use the widget below to live the relationship. Switch between curve types, then bend each one — drag its handles and watch the shape respond while the eccentricity readout updates in real time.
Watch the eccentricity readout. Push an ellipse's \( a \) and \( b \) apart and \( e \) climbs toward \( 1 \) — the oval stretches longer and flatter. The parabola sits exactly at \( e = 1 \), the knife-edge between closed and open. The hyperbola lives beyond, at \( e > 1 \). One number, sliding from \( 0 \) upward, walks you through every conic in order.
The ellipse
Centre an ellipse at the origin with its long axis horizontal, and it obeys the clean standard form
\[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1, \qquad a > b > 0. \]Here \( a \) is the semi-major axis (half the long width) and \( b \) the semi-minor axis (half the short width). Inside sit two special points, the foci, at \( (\pm c, 0) \), where
\[ c^2 = a^2 - b^2. \]What makes those two points special is the ellipse's defining property: for every point on the curve, the sum of the distances to the two foci is constant, equal to \( 2a \). This is the famous two pins and a loop of string construction — pin both ends of a loop, pull it taut with a pencil, and trace; the pencil draws a perfect ellipse because the string length never changes. The eccentricity is
\[ e = \frac{c}{a}, \]which is \( 0 \) when the foci coincide at the centre (a circle) and approaches \( 1 \) as they slide toward the ends. This focus-pair property is why a whispering gallery works — a whisper at one focus gathers back together at the other — and why planets sweep out elliptical orbits with the Sun parked at one focus.
The parabola
The parabola is the borderline case, \( e = 1 \), where focus and directrix carry equal weight. With its vertex at the origin and opening upward, it has the tidy form
\[ y = \frac{x^2}{4p}, \]with its single focus at \( (0, p) \) and its directrix the horizontal line \( y = -p \). Every point on the curve is equidistant from the focus and the directrix — that perfect balance is the whole geometry of the parabola in one sentence.
This is no idle curiosity. The parabola has a stunning reflective property: any ray arriving parallel to the axis bounces off the curve and passes straight through the focus. Run it backwards and a source at the focus throws out a perfectly parallel beam. That is precisely why satellite dishes, radio telescopes and car headlights are shaped as parabolas — to gather faint signals to a single point, or to cast light into a tight beam.
The hyperbola
Tilt the slice past the parabola's angle and the curve splits in two. Centred at the origin and opening left-and-right, the hyperbola's standard form swaps a plus for a minus:
\[ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1. \]It has two opening branches, with foci at \( (\pm c, 0) \) — but now
\[ c^2 = a^2 + b^2, \]so the foci sit outside the branches. Far from the centre, each branch hugs a pair of straight lines it never quite touches, the asymptotes
\[ y = \pm \frac{b}{a}\,x, \]which the branches race toward as \( x \) grows. The defining property mirrors the ellipse's but trades sum for difference: for every point on the hyperbola, the difference of the distances to the two foci is constant, equal to \( 2a \). And the eccentricity
\[ e = \frac{c}{a} > 1 \]always exceeds one, the signature of an open curve.
- This is an ellipse in standard form. Read off the squared denominators: \( a^2 = 25 \) and \( b^2 = 9 \), so \( a = 5 \) and \( b = 3 \). Since \( a > b \), the major axis is horizontal.
- For an ellipse the foci use \( c^2 = a^2 - b^2 \): \( c = \sqrt{25 - 9} = \sqrt{16} = 4 \).
- The foci therefore sit at \( (\pm 4, 0) \) on the major axis.
- The eccentricity is \( e = \dfrac{c}{a} = \dfrac{4}{5} = 0.8 \).
Foci \( (\pm 4, 0) \) and \( e = \mathbf{0.8} \) — close enough to \( 1 \) that this ellipse is noticeably stretched, not nearly round.
- The minus sign marks this as a hyperbola opening left and right. Read off \( a^2 = 9 \) and \( b^2 = 16 \), so \( a = 3 \) and \( b = 4 \).
- For a hyperbola the foci use \( c^2 = a^2 + b^2 \): \( c = \sqrt{9 + 16} = \sqrt{25} = 5 \), giving foci at \( (\pm 5, 0) \).
- The asymptotes are \( y = \pm \dfrac{b}{a}x = \pm \dfrac{4}{3}x \) — the diagonal lines the branches chase.
- The eccentricity is \( e = \dfrac{c}{a} = \dfrac{5}{3} \approx 1.67 \), comfortably above \( 1 \).
Two branches opening sideways, foci at \( (\pm 5, 0) \), asymptotes \( y = \pm \tfrac{4}{3}x \), and \( e = \mathbf{\tfrac{5}{3} \approx 1.67} \).
Practice
Try each one yourself, then reveal the full solution.
1. Find the focus and directrix of \( y = \dfrac{x^2}{8} \).
Compare with the standard parabola \( y = \dfrac{x^2}{4p} \). Matching denominators, \( 4p = 8 \), so \( p = 2 \).
The focus sits at \( (0, p) = (0, 2) \), and the directrix is the line \( y = -p = -2 \).
Focus \( \mathbf{(0, 2)} \), directrix \( \mathbf{y = -2} \). Every point on the curve is the same distance from that point as from that line.
2. An ellipse has \( a = 13 \) and \( b = 5 \). Find its foci and eccentricity.
For an ellipse, \( c^2 = a^2 - b^2 \), so \( c = \sqrt{169 - 25} = \sqrt{144} = 12 \).
The foci lie at \( (\pm 12, 0) \) along the major axis.
The eccentricity is \( e = \dfrac{c}{a} = \dfrac{12}{13} \) — close to \( 1 \), so this is a long, flattened ellipse.
3. Two ellipses have eccentricities \( e = 0.2 \) and \( e = 0.9 \). Which is nearly circular?
Eccentricity measures how far an ellipse departs from a circle. A circle has \( e = 0 \), so the closer \( e \) is to zero, the rounder the shape.
The ellipse with \( e = \mathbf{0.2} \) is the nearly circular one. The \( e = 0.9 \) ellipse is long and flattened, with its foci pushed far out toward the ends.