Functions
AI Assistant

Overview of Conic Sections

Circle, ellipse, parabola, hyperbola — four curves from one cone

Every conic section comes from slicing a cone with a plane at different angles. Cut straight across → circle. Tilt the plane → ellipse. Cut parallel to the side → parabola. Cut steep enough to hit both halves → hyperbola.

Each has a standard equation and distinct geometric properties: the circle has constant radius, the ellipse has two foci, the parabola has a focus and directrix, and the hyperbola has asymptotes.

In this lesson you'll see all four on one graph and compare their equations and shapes.

Graph

FAQ

What are conic sections?
Conic sections are curves formed by the intersection of a plane with a double-napped cone. The four types are: circle, ellipse, parabola, and hyperbola. They appear everywhere — planetary orbits (ellipses), satellite dishes (parabolas), and cooling towers (hyperbolas).
What is the equation of each conic section?
Circle: x^2 + y^2 = r^2. Ellipse: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1. Parabola: y = ax^2. Hyperbola: \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1.
How are ellipses and circles related?
A circle is a special case of an ellipse where both semi-axes are equal (a = b = r). As you stretch an ellipse in one direction, it becomes more elongated and its eccentricity increases from 0 (circle) toward 1.
What makes a hyperbola different from an ellipse?
An ellipse has a plus sign between the terms: x²/a² + y²/b² = 1. A hyperbola has a minus sign: x²/a² − y²/b² = 1. The ellipse is a closed curve; the hyperbola has two separate branches that extend to infinity.

Related Topics

The EllipseThe HyperbolaQuadratic Equations