Unit-I: Sets and Functions
Unit-II: Algebra
Unit-III: Coordinate Geometry
Unit-IV: Calculus
Unit-V: Mathematical Reasoning
Unit-VI: Statistics and Probability

Introduction to Conic Sections

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:

N.B :

 V -vertex

 l =axis of cone

 m= generator (rotating line )

\to 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.

\to The fixed point is called Focus.

\to The fixed straight line is called the Directrix.

 \to The constant ratio is called the Eccentricity, denoted by  e .

\to The line passing through the focus and perpendicular to the direction is called the Axis.

 \to The point of intersection of a conic with its axis is called the vertex.

Note:

\to The circle, where the cone is cut at right- angles to its axis.

\to The ellipse, where the cone is cut at an oblique angle shallower than a generator.

\to The Parabola, where the cone is cut parallel to a generator.

 \toThe hyperbola, where a double-napped cone is cut at an angle steeper than a generator.

Eccentricity (e):

 

\to If  0 < e < 1 , then the curve is an ellipse.

\to If  e=1 , then the curve is a parabola.

\to If  e > 1, then the curve is a hyperbola.

 

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