Conic Sections
    Conics, a Family of Similarly Shaped Curves - Properties of Conics
      Dandelin's Spheres - proof of conic sections focal properties
         Proof that conic section curve is the parabola
      Conics - a family of similarly shaped curves
Dandelin's spheres proof that conic section curve is the parabola
When the cutting plane is parallel to any generator of one of the cones then we can insert only one sphere into the cone which will touch the plane at the point F and the cone surface at the circle k.
Arbitrary chosen generating line g intersects the circle k at a point M, and the intersection curve p at a point P. The point P lies on the circle k' which is parallel with the plane K as shows the down figure.
By rotating the generator g around the vertex V, the point P will move along the intersection curve. 
   While the generator approaches position to be parallel to the plane E, the point P will move far away from F. That shows the basic property of the parabola that the line at infinity is a tangent.
  The segments, PF and PM belong to tangents drawn from P to the sphere
so,  PM  = PF.
  Since planes of the circles, k and k'   are parallel to each other and               perpendicular to the section through      the cone axis, and as the plane E is     parallel to the slanting edge VB, then    the intersection d, of planes E and K,   is also perpendicular to the section       through the cone axis.                        
 Thus, the perpendicular PN from P to  the line d,
PN = BA = PM  or  PF = PN.
Therefore, for any point P on the intersection curve the distance from the fixed point F is the same as it is from the fixed line d, it proves that the intersection curve is the parabola.
Conics - a family of similarly shaped curves
A conic is the set of points P in a plane whose distances from a fixed point F (the focus) and a fixed line d (the directrix), are in a constant ratio. This ratio named the eccentricity e determines the shape of the curve.
We can see that conics represent a family of similarly shaped curves if we write their equations in vertex form.
Recall the method we used to transform equations of the ellipse and the hyperbola from standard to 
vertex form. We placed the vertex of the curve at the origin translating its graph. 
  Thus, obtained are their vertex equations;
y2 = 2px - (p/a)x2
- the ellipse and the circle
(for the circle  p = a = r)
y2 = 2px 
- the parabola
y2 = 2px + (p/a)x2
- the hyperbola
   Using geometric interpretation of these equations we compare the area of the square y2, formed by the ordinate of a point P(x, y), with the area of the rectangle 2p · x, whose one side is the abscissa x of the point P and the parameter 2p other side, it follows that
    - for the ellipse the area of the square is smaller, than the area of the rectangle,
    - for the parabola is equal,
    - for the hyperbola the area of the square is greater than the area of the rectangle.
The names of curves were given as a result of the above relations, so;
    - the word “ellipse” (elleipyis) in Greek means “deficiency,"
    - the word “parabola” (parabolh) means “equality” and 
    - the word  “hyperbola” (uperbolh) means “excess.”
In the given vertex equations we can make following substitutions for; 
  - the ellipse
  - the circle      p = a = b = r   =>    e = 0
  - the parabola     e = 1
  - the hyperbola
Thus, the equation of conics in vertex form is  y2 = 2px - (1  - e2)x2.  
The values of e define the curve the conic section makes, such that for
                           e = 0    - a circle,
                     0 < e < 1    - an ellipse,
                           e = 1    - a parabola,
                           e > 1    - a hyperbola,
as shows the above figure.
College algebra contents E
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