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Sections of a Cone

Consider a fixed vertical line l. Let another line m intersect l at the point V inclined at an angle α . The surface generated by revolving the line m about the line l in such a way that the angle α remains constant, is a double - napped right circular hollow cone.

l - axis of the cone
m - generator of cone
V - vetex of cone


The section obtained when a plane intersects a cone is called a conic section.
Let the angle made by the plane with the vertical axis of the cone be β. For different values of β we get different types of conic sections as illustrated in figure given below.
The section is a circle if β = 90°

The section is an ellipse if α < β < 90°

The section is a parabola if α = β

The section is a hyperbola if 0 β < α


In case the cutting plane passes through the vertex, the section is a 'degenerated conic'.
When α < β 90° then the degenerated conic section is a point which is nothing but a degenerated circle or ellipse.


When α = β then the degenerated conic section is a single straight line which is nothing but a degenerated parabola.


When 0 β < α then the degenerated conic section is a pair of straight lines which is nothing but a degenerated hyperbola.
There are many applications associated with conic sections. The planets orbit the sun in elliptical orbits. Many of the comets that come in contract with the gravitational field surrounding the earth travel in parabolic or hyperbolic paths. Flash light mirrors have elliptical or parabolic shapes because of the way surfaces with those shapes reflect light. Galileo discovered that if a projectile is fired horizontally from the top of a tower, then its path would be a parabola.

Conic sections were studied extensively by the ancient Greeks, who discovered properties that enable us to state their definitions in terms of points and lines.

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