Mathematics Class XI

Unit-I: Sets and Functions
Chapter 1: Sets
Unit-II: Algebra
Chapter 5: Binomial Theorem
Chapter 6: Sequence and Series
Unit-III: Coordinate Geometry
Chapter 1: Straight Lines
Chapter 2: Conic Sections
Unit-IV: Calculus
Unit-V: Mathematical Reasoning
Unit-VI: Statistics and Probability
Chapter 1: Statistics
Chapter 2: 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 :

-vertex

=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.

Note:

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.

Eccentricity (e):

If , then the curve is an ellipse.

If , then the curve is a parabola.

If , then the curve is a hyperbola.

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