Circle, parabola, ellipse & hyperbola
Four curves, one family. A conic is just what you get when a plane slices a double cone — but you don't need the cone to see it. Pick a shape below and drag its parameters. The curve, its foci, its eccentricity and its directrix all update live, straight from the standard equation.
The ideas you're seeing
What makes a conic a conic
All four curves come from slicing a double cone with a plane at different angles: a circle (plane perpendicular to the axis), an ellipse (tilted plane, still crossing only one nappe), a parabola (plane parallel to the cone's slant), and a hyperbola (plane crossing both nappes).
Eccentricity tells you which conic
e = 0 is a circle, 0 < e < 1 is an ellipse, e = 1 is a parabola, e > 1 is a hyperbola — a single number that captures how "stretched" or "open" the curve is.
Foci and the defining property
Every point on an ellipse has a constant sum of distances to the two foci; every point on a hyperbola has a constant difference of distances to its two foci. This focal-distance property is literally how gardeners draw ellipses with a loop of string around two pegs.
The parabola's focus-directrix property
Every point on a parabola is equidistant from the focus and the directrix — this single property, not "y = x²", is the actual geometric definition. It's why parabolic mirrors and satellite dishes focus parallel rays to a single point.
Part of the Conic Sections chapter — read the notes, grab the formula sheet and take the quiz. One of Priodemy for School, free with every EduSuite school.
Four curves, one family
Why they are called sections
The circle, parabola, ellipse and hyperbola are usually met as four unrelated equations to be memorised separately. They are in fact a single family: take a double cone and slice it with a plane, and the shape of the cut depends only on the angle of the slice. Cut horizontally and you get a circle. Tilt slightly and it stretches into an ellipse. Tilt until the plane is parallel to the cone's slant and the curve opens out into a parabola. Tilt further so the plane meets both halves of the cone and you get the two branches of a hyperbola.
That is why they share so much algebraic structure, and why one parameter can carry you continuously from one to the next.
Eccentricity is that parameter
Every conic can be defined by a single rule: the set of points whose distance from a fixed focus divided by the distance from a fixed line, the directrix, is a constant. That constant is the eccentricity e, and it alone decides the shape.
When e = 0 the curve is a circle. For 0 < e < 1 it is an ellipse, becoming visibly more elongated as e climbs. At exactly e = 1 it is a parabola — the knife-edge case. For e > 1 it is a hyperbola. Slide e here and watch one curve morph continuously into the next; the parabola is not a separate object but the boundary between closed and open curves.
The focal properties, and why they are useful
Each conic has a distance property that defines it. For an ellipse, the sum of distances to the two foci is constant — the reason a loop of string round two pins draws one. For a hyperbola, the difference of those distances is constant instead. For a parabola, every point is equidistant from focus and directrix.
These are not abstractions. The parabola's property means all rays arriving parallel to the axis reflect through the focus, which is why satellite dishes, torch reflectors and radio telescopes are parabolic. The ellipse's property means anything emitted at one focus converges on the other, used in whispering galleries and in lithotripsy to focus shock waves on a kidney stone. Planetary orbits are ellipses with the Sun at one focus, which is Kepler's first law.
Mistakes that cost marks
Assuming a is always the larger denominator. In x²/a² + y²/b² = 1 the major axis lies along whichever denominator is bigger. If b > a the ellipse is taller than it is wide, and the foci sit on the y-axis, not the x-axis.
Using the wrong focal relationship. For an ellipse b² = a²(1 − e²), so c² = a² − b². For a hyperbola the sign flips: c² = a² + b². Carrying the ellipse formula into a hyperbola question is a standard trap.
Failing to complete the square. An equation with x and y terms present describes a shifted conic. Complete the square in both variables to find the true centre before identifying the curve or reading off a and b.