When a plane cuts a cone, we get various types of plane sections depending upon the positions of the plane. These curves are known as parabola, ellipse, hyperbola, and also a circle.
The parabola, ellipse, and hyperbola are mainly called the Conic sections. They have well-defined directrices, which will be defined by their respective properties. But the directrices of a circle are not defined. For this reason, though a circle comes as a section of a cone when the axis of the cone is normal to the plane cutting it. A circle is, however, called a conic section of the fourth type.
Diagrammatic Representation of Conic Sections:
=axis of cone
= generator (rotating line )
The vertex separates two cones in two parts, called Nappe.
Definition of Conic Sections:
A Conic section or Conic is the locus of a point which moves in a plane so that its distance from a fixed point is in a constants ratio to its perpendicular distance from a fixed straight line.
The fixed point is called Focus.
The fixed straight line is called the Directrix.
The constant ratio is called the Eccentricity, denoted by .
The line passing through the focus and perpendicular to the direction is called the Axis.
The point of intersection of a conic with its axis is called the vertex.
The circle, where the cone is cut at right- angles to its axis.
The ellipse, where the cone is cut at an oblique angle shallower than a generator.
The Parabola, where the cone is cut parallel to a generator.
The hyperbola, where a double-napped cone is cut at an angle steeper than a generator.
If , then the curve is an ellipse.
If , then the curve is a parabola.
If , then the curve is a hyperbola.