Conic Sections

Conic Sections

Conic sections are curves that arise from the intersection of a plane with a double-napped right circular cone. Depending on the angle at which the plane slices through the cone, different shapes are formed.

The four primary conic sections are: Circle, Ellipse, Parabola, and Hyperbola

The following figures show some examples of the four conic sections. Scroll down the page for more examples.

 

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How to form Conic Sections

Imagine a double cone (two cones joined at their vertices, or apex). A plane slices through this double cone:

1. Circle: When the plane cuts perpendicular to the axis of the cone. It’s a closed, perfectly round curve.

2. Ellipse: When the plane cuts through one nappe of the cone at an angle, but not perpendicular to the axis, and not parallel to the side of the cone. It’s an oval-shaped closed curve. A circle is a special case of an ellipse where the plane is perfectly perpendicular to the axis.

3. Parabola: When the plane cuts through one nappe of the cone parallel to a generator (the slant edge) of the cone. It’s an open, U-shaped curve.

4. Hyperbola: When the plane cuts through both nappes of the cone, parallel to its axis, or at an angle such that it intersects both cones. It consists of two separate, open curves (two branches)

Videos

Introduction To Conic Sections

By definition, a conic section is a curve obtained by intersecting a cone with a plane.

In Algebra II, we work with four main types of conic sections: circles, parabolas, ellipses and hyperbolas. Each of these conic sections has different characteristics and formulas that help us solve various types of problems.

This video shows how you can generate a circle, ellipse, parabola, and hyperbola by intersecting a cone with a plane. It is the first of several videos on the conic sections.

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How To Create The Different Conic Sections?

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Introduction To Conic Sections

What are conic sections and why are they called “conic sections”?

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Free Math Lessons: Introduction to Conic Sections

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