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 hyperbola
         Proof that conic section curve is the parabola
Dandelin Spheres - proof of conic sections focal properties
Proof that conic section curve is the hyperbola
When the intersecting plane is inclined to the vertical axis at a smaller angle than does the generator of the 
cone, the plane cuts both cones creating the               hyperbola h which therefore consists of two disjoining  branches as shows the right figure.                            
  Inscribed spheres touch the plane on the same side   at points F1 and F2 and the cone surface at circles     k1and k2.                                                                
  The generator g intersects the circles k1and k2, at    points, M and N, and the intersection curve at the       point P.                                                                  
  By rotating the generator g around the vertex V by    
360, the point P will move around and trace both      
branches of the hyperbola.                                        
  While rotating, the generator will coincide with the      plane two times and then will have common points       with the curve only at infinity.                                    
  As the line segments, PF1and PM are the tangents  segments drawn from P to the upper sphere, and the   segments PF2 and PN are the tangents segments     drawn to the lower sphere, then                                 
PM = PF and  PN = PF2.
  Since the planes of circles k1 and k2, are parallel,     then are all generating segments from k1to k2 of equal length, so                                                                
MN = PM - PN  or  PF1 - PF2  is constant.  
  Thus, the intersection curve is the locus of points in  
the plane for which difference of distances from the      two fixed points
F1 and F2, is constant, i.e., the curve is the hyperbola.                                                       
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.
Pre-calculus contents H
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