CONIC SECTIONS

 The greek mathematician Archimedes and Apollonius studied the curve named conic sections . These curve are intersection of a plane with right circular cone. Conic section have a wide range of applications such as planetary motions, in desigens of telescope and antennas, eflection in flash , automobile headlights, construction of bridge , navigation, projectiles etc.

Definition of conic section:

    A conic section or conic can be defined as the locus of the P in a plane such that the ratio of the distance of P from a fixed point to its distance from a fixed line is constant e known as eccentricity. 

  The fixed point is callled the focus of the conic section, denoted by S . The fixed straight line is called the directrix of conic section, denoted by  d.

If S is the focus , P is any point on the conic section and segment PM is the length of perpandicular from P on the derectrix, then by definition SP/PM = constant.

            

The nature of the conic section depends upon the value of e .

i)  If e = 1 , then conic section is called parabolla.

ii)  If 0 < e < 1 , the conic sction is called an ellipse.

iii) If e > 1, the conic section is callled as hyperbolla.

iv) If e = 0 , the conic section is callled as circle.

So, eccentricity is a measure of the deviation of the ellipse from being circular. Suppose, the angle formed between the surface of the cone and its axis is β and the angle formed between the cutting plane and the axis is α, the eccentricity is;

e = cos α/cos β

Some useful terms of conic section :

1) Axis :  A line about which a conic section is symmetric is callled an axis of the conic section.

2) Vertex : The point of intersection of a conic section with its axis of symmetric is called a vertes.

3) Focal Distance : The distance of a point on a conic section from the focus is callled the focal distance of the point.

4) Focal cord : A chord of a conic section passing through its focus is callled a focal chord .

5) Latus-Rectum:  A focal chord of a conic section which is perpandicular to the axis of symmetric is called as the latus-rectum.

6) Centre of a conic : The point which bisects every chord of the conic passing through it, is callled as the centre of the conic.

7) Double Ordinate :  A chord passing through any point on the conic and perpandicular to the axis is callled as double ordinate.

There are four types of conic section 

  • Circle 
  • ellips 
  • Parabolla
  • Hyperbola

Parameters of Conic

  • Principal Axis: Line joining the two focal points or foci of ellipse or hyperbola. Its midpoint is the centre of the curve.
  • Linear Eccentricity: Distance between the focus and centre of a section.
  • Latus Rectum: A chord of section parallel to directrix, which passes through a focus.
  • Focal Parameter: Distance from focus to the corresponding directrix.
  • Major axis: Chord joining the two vertices. It is the longest chord of an ellipse.
  • Minor axis: Shortest chord of an ellipse.
Ellipse :

     The ellipse is the intersection of double nappped cone with an oblique plane . If α < β < 90o, the conic section so formed is an ellipse as shown in the figure below.
     

Standard equation of ellipse :  
    
Parabolla :

   A parabolla is the locus of the point in plane equidistant from a fixed point and a fixed line in that plane. The fixed point is called the focus and the fixed straight line is called the directrix.
If α=β, the conic section formed is a parabola (represented by the orange curve) as shown below.

Some standard form of parabolla :

Standard equation of parabolla :
  Eqution of the parabolla in the standard form y2 = 4ax.

Hyperbola :
 
The hyperbola is the intersection of double napped cone with plane parallel to the axis .  The hyperbolla is the locus of a point in a plane which moves so that its distance from a fixed point bears a constant ratio e (e > 1) to its distance from a fixed line. The fixed point is called as focus and the fixed line is callled the directrix d.
If 0≤β<α, then the plane intersects both nappes and the conic section so formed is known as a hyperbola 

Standard form of hyperbola :
  
How To Construct Conic Section :

i)  The plane is perpandicular to the axis and does contain vertex, the intersection is a circle .
ii)  The plane is parallel to one position of the generator but does not pass through the vertex, we get a         parabola.
ii) The plane is oblique to the axis and not parallel to the generator we get an ellipse .
iv) If a double cone is cut by a plane parallel to axis , we get parts of the curve at two ends called                 hyperbola.
v) A plane containing a generator and tangent to the cone, intersects the cone in that generator. we get        pair to straight lines .

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