Conic sections - Dandelin's Spheres - proof of conic sections focal properties

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's 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 F₂and 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 PF₂and 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 k₂of 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 kat 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. 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.

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 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.

Intermediate algebra contents