Conic Sections and Eccentricity(e)

Let us examine the process of Limit e ---> 1of a conic

 The X shaped lines that define the surface of the cone are called generators and the vertical line passing through the intersection of the generator is called axis 
 
We have conic sections formed when a double right circular cone is intersected by a plane.  
General equation of a conic is  b²x²+a²y² = a²b²

where, b² = a²|1-e²|

When e = 0 , we get the conic section- 'Circle'
Note: Here the plane cuts the double cone at right angle with the axis of the cone

  When e < 1 , we get the conic section- 'Ellipse'

Note: Here the plane cuts the double cone at an inclination with the axis of the cone

 When e = 1 , we get the conic section- 'Parabola' 
Note: Here the plane lies parallel to the generator of the double cone

 When e > 1 , we get the conic section- 'Hyperbola'
Note : Here the plane cuts bottom and top cones of the double cone

 So as Eccentricity(e) varies the conic section varies as shown below

Key Note: Can we conclude the process in the following form

when e = 0 we have a perfectly symmetrical figure which is a circle. 

As e moves from 0 to 1, the circle gets deformed. As e approaches 1 the deformation becomes deeper and deeper and as e equals 1 there is an explosion instantaneously and the conic section becomes a parabola. Further as e moves from 1 instaneously another explosion results into a hyperbola.