
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
k_{1}
and the plane at a point F_{1}.
Lower sphere touches the cone surface in a circle
k_{2
} and the plane at a
point
F_{2}.

Arbitrary chosen generating line g
intersects the
circle
k_{1
}at a point
M, the circle
k_{2}
at a point
N
and
the intersection curve e
at a point
P.

We see that points,
M
and F_{1}
are the tangency
points of the upper sphere and points, N
and F_{2}
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_{1 }
and PN =
PF_{2}.

And since planes of
circles k_{1}and
k_{2},
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 F_{1}and
F_{2}, 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
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.













College
algebra contents E 



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