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Conic Sections

  • Conic sections 

Conic sections or conics are the curves that are obtained by intersecting a plane with a double-napped right circular cone. Circles, ellipses, parabolas and hyperbolas are examples of conic sections.

A double-napped cone can be obtained by rotating a line (let us say m) about a fixed vertical line (let us say l).

 

Here, the fixed line l is called the axis of the cone and m is called the generator of the cone. The intersection (V) of l and m is called the vertex of the cone.

Different conics formed by intersecting a plane and a double-napped cone:

If θ1is the angle between the axis and the generator and θ2is the angle between the plane and the axis, then, for different conditions of θ1and θ2, we get different conics, which are described with the help of a table as shown below.

 

Condition

Conic Formed

Figure

θ2 = 90° (Only one nappe of the cone is entirely cut by the plane)

A circle

θ1< θ2< 90° (Only one nappe of the cone is entirely cut by the plane)

An ellipse

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