
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
F_{1}
and F_{2
}and the cone surface at circles
k_{1}and k_{2}.

The generator
g
intersects the circles k_{1}and k_{2}, 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, PF_{1}and
PM are the tangents
segments drawn from P
to the upper sphere, and the segments PF_{2
}and
PN
are the tangents segments drawn to the lower sphere, then

PM =
PF_{1 }
and PN =
PF_{2}.

Since the planes of
circles k_{1}
and k_{2}, are parallel,
then are all generating segments from
k_{1}to
k_{2
}of equal
length,
so

MN =
PM 
PN
or PF_{1}

PF_{2}
is constant.

Thus, the intersection curve is the locus of points in
the plane for which difference of distances from the
two fixed
points F_{1}
and F_{2}, 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.









Precalculus contents
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