# Sections of a Cone

Consider a fixed vertical line*l*. Let another line

*m*intersect

*l*at the point V inclined at an angle Î± . The surface generated by revolving the line

*m*about the line

*l*in such a way that the angle Î± 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 Î². For different values of Î² we get different types of conic sections as illustrated in figure given below. |

*circle*if Î² = 90Â°

The section is an

*ellipse*if Î± < Î² < 90Â°

The section is a

*parabola*if Î± = Î²

The section is a hyperbola if 0 â‰¤ Î² < Î±

In case the cutting plane passes through the vertex, the section is a 'degenerated conic'.When Î± < Î² â‰¤ 90Â° then the degenerated conic section is a point which is nothing but a degenerated circle or ellipse. |

When Î± = Î² then the degenerated conic section is a single straight line which is nothing but a degenerated parabola. |

When 0 â‰¤ Î² < Î± then the degenerated conic section is a pair of straight lines which is nothing but a degenerated hyperbola. |

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.