Conic sections

8 min read · Updated 20 October 2026

Slice a double cone with a flat plane and you get one of four curves: a circle, an ellipse, a parabola or a hyperbola. Each has a standard equation, and every second-degree equation in x and y is one of them, or a degenerate case.

Slicing a cone

Tilt the cutting plane gradually. Perpendicular to the cone’s axis it gives a circle. Tilted a little, an ellipse. Parallel to the side of the cone, a parabola. Steeper still, so it cuts both halves of the double cone, a hyperbola with two branches. Apollonius of Perga named and studied all four around 200 BC.

The standard equations

Conic	Standard form (centre or vertex at the origin)
Circle	x² + y² = r²
Ellipse	x²/a² + y²/b² = 1
Parabola	y = x²/(4p), or x = y²/(4p)
Hyperbola	x²/a² − y²/b² = 1

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A circle, an ellipse, a parabola and a hyperbola, all typed exactly as written.

Foci and what makes each curve special

Ellipse: the set of points whose distances to two foci add up to a constant. With a > b the foci are at (±c, 0) with c² = a² − b². Planets orbit the Sun along ellipses with the Sun at one focus.
Parabola: points equally far from a focus (0, p) and a directrix y = −p. Rays parallel to the axis reflect through the focus, which is how satellite dishes work.
Hyperbola: points whose distances to two foci differ by a constant, with c² = a² + b². Its asymptotes are y = ±(b/a)x.
Circle: an ellipse whose two foci coincide at the centre.

Identifying a conic from its equation

For Ax² + Cy² + Dx + Ey + F = 0 (no xy term):

A = C: a circle.
A and C have the same sign but differ: an ellipse.
A and C have opposite signs: a hyperbola.
Exactly one of A and C is zero: a parabola.

With an xy term, Ax² + Bxy + Cy² + …, use the discriminant B² − 4AC: negative for an ellipse (or circle), zero for a parabola, positive for a hyperbola. The xy term rotates the curve.

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x²+xy+y²=3

has B² − 4AC = −3, a rotated ellipse;

xy=2

has B² − 4AC = 1, a rotated hyperbola.

Graphing conics in the calculator

Type any conic exactly as written, rotated or not. Because the calculator plots implicit equations directly, there is no need to solve for y or split the curve into halves. Use sliders for a and b, x^2/a^2 + y^2/b^2 = 1, to watch an ellipse become a circle when a = b.

Practice

A new problem every time. Answer in any form you like: fractions, decimals, √ and π all work, and equivalent equations count as right.

Problem 1

What kind of conic is y²/9 − x²/1 = 1?

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Answer: Hyperbola

x² and y² have opposite signs: a hyperbola.

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Frequently asked questions

What are the four conic sections?

The circle, ellipse, parabola and hyperbola: the curves made by slicing a double cone with a plane.

How can I tell which conic an equation is?

Compare the x² and y² coefficients: equal means a circle, same sign means an ellipse, opposite signs a hyperbola, and only one squared term a parabola. With an xy term, use B² − 4AC.

What is the focus of a parabola?

For y = x²/(4p) it is the point (0, p). Every point on the parabola is the same distance from the focus as from the directrix y = −p.

What are the asymptotes of a hyperbola?

For x²/a² − y²/b² = 1 they are the lines y = ±(b/a)x.