Conic Sections
Conics, a Family of Similarly Shaped Curves - Properties of Conics Conics, a Family of Similarly Shaped Curves - Properties of Conics Proof that conic section curve is the ellipse

Proof that conic section curve is the parabola Conics - a family of similarly shaped curves
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 = PF1  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 is 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 = PF1  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.
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.   Conic sections contents 