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 ellipse
         Proof that conic section curve is the hyperbola
Conics, a family of similarly shaped curves - properties of conics
  By intersecting either of the two right circular conical surfaces (nappes) with the plane perpendicular to the axis of the cone the resulting intersection is a circle c, as is shown in the figure.
  When the cutting plane is inclined to the axis of the cone at a greater angle than that made by the generating segment or generator (the slanting edge of the cone), i.e., when the plane cuts all generators of a single cone, the resulting curve is the ellipse e.
Thus, the circle is a special case of the ellipse in which the plane is perpendicular to the axis of the cone.
   If the cutting plane is parallel to any generator of one of the cones, then the intersection curve is the parabola p.
  When the cutting plane is inclined to the axis at a smaller angle than the generator of the cone, i.e., if the intersecting plane cuts both cones the hyperbola h is generated.
Dandelin Spheres - proof of conic sections focal properties
Proof that conic section curve is the ellipse
In the case when the plane E intersects all generators of the cone, as in down figure, it is possible to inscribe two spheres which will touch the conical surface and the plane.
Upper sphere touches the cone surface in a circle k1 and the plane at a point F1. Lower sphere touches the cone surface in a circle k2 and the plane at a point F2.
  Arbitrary chosen generating line g intersects the     
circle
k1 at a point M, the circle k2 at a point N and  
the intersection curve
e at a point P.                        
  We see that points, M and F1 are the tangency     
points of the upper sphere and points,
N and F2 the  
tangency points of the lower sphere of the tangents  
drawn from the point
P exterior to the spheres.         
  Since the segments of tangents from a point           exterior to sphere to the points of contact, are equal  
PM = PF and  PN = PF2.
And since planes of circles k1and k2, are parallel,     then are all corresponding generating segments        equal                                                                    
MN = PM + PN  is constant.
  Thus, the intersection curve is the locus of points    in the plane for which sum of distances from the two  fixed points F1and F2, is constant, i.e., the curve is   the ellipse.                                                            
The proof due to the French/Belgian mathematician Germinal Dandelin (1794 1847).
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.                                                       
College algebra contents E
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