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Sections of a Cone

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Sections of a Cone - Lesson Summary

Cone

Consider a fixed vertical line l.

Consider another line m that intersects line l at a fixed point V.

Let the angle at which line m is inclined to line l be ∞.

If line m rotates about point V such that angle ∞ remains constant, it generates a double cone. Such a cone is called a double right-circular hollow cone.

The fixed vertical line passing through the centre of the cone is called its axis. The rotating line m that generates the surface of the cone is called the generator. The fixed point at which the generator cuts the axis is called the vertex of the cone.

The cone is divided into two parts by the vertex. The upper part is called the upper nappe and the lower one is called the lower nappe.

Conic sections

When a plane intersects a cone, it cuts a section from the cone. This section is called a conic section or a conic.

Different kinds of conic sections of different shapes are obtained depending on the position of the intersecting plane with respect to the cone and the angle it makes with the vertical axis of the cone.

Consider a plane that intersects a cone at any position other than the vertex and makes an angle b with the axis of the cone.

If the plane intersects one nappe of the cone at right angle to the vertical axis, the resulting conic section is a circle.

If the plane intersects one nappe of the cone such that angle b is greater than angle a, but less than 90°, the resulting conic section is an ellipse.

If the plane intersects one nappe of the cone such that angle b is equal to angle a, the resulting conic section is a parabola.

If the plane intersects the cone such that angle beta is less than angle alpha and greater than or equal to zero, the plane cuts across both nappes of the cone, and the resulting conic section is a hyperbola.

In the formation of conic sections in the form of a circle, ellipse or parabola, the plane intersects only one nappe of the cone. However, in the formation of a conic section in the form of a hyperbola, the plane intersects both the nappes of the cone.

In the formation of these conic sections, the intersecting plane does not pass through the vertex of the cone.

Practical applications of conic sections:

Historically, the circle has provided momentum to our economic and industrial development.

The ellipse is used extensively for studying planetary motion and making advanced reflectors for automobile headlights.

Parabolas are used to study the trajectories of projectiles.

Hyperbolas are used to study sonic booms created by supersonic aircraft, and to study the propagation of waves and movement of sub-atomic particles.

The study of these conic sections finds many practical applications in different branches of science.

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