This document discusses properties of hyperbolas using Dandelin spheres. It shows that for any point P on a hyperbola, the distance PF to the focus F is equal to the distance PC to the directrix. This property does not change as P moves along the hyperbola. The document also derives formulas for the external and internal tangents of two circles based on their radii and distance between centers. These concepts are then applied to define the eccentricity of a hyperbola in terms of the radii of the Dandelin spheres. The method shown can also be used to describe other conic sections like parabolas.

1. About hyperbola, an example of
learning shifts on Internet
Danut Dragoi, PhD
California
01/28/2017

2. 2
Dandelin spheres for hyperbola

3. 3
Property of hyperbola
For any point P on hyperbola on
upper branch, PF=PC, as tangents to
the small sphere and C is on small
circle, PF'=PC', C' on the larger circle
and PC' goes through the vertex V of
the cones, then PF'-PF=PC'-
PC=CC'=constant.
-CC' is the internal tangent:
CC'=SQRT(d2
-(r2
+r1
)2
)
and FF' is the external tangent (see
the derivations on slides #4 and #5):
FF'=SQRT(d2
-(r1
-r2
)2
)

4. External tangent on two circles,
given radii r1
, r2
, and O1
O2
=d
L=SQRT(d2
-(r1
-r2
)2
)

5. Internal tangent on two circles,
given radii r1
, r2
, and O1
O2
=d
L=SQRT(d2
-(r2
+r1
)2
)

6. 6
Position of the directrix
For any point P on hyperbola, upper branch,
consider the triangles PQS and PRT, where
PQ and PR are perpendiculars from P on lines
d and d' (d=Plane α ∩ plane π, d'=Plane α' ∩
plane π), and PS and PT are lines that go
through the vertex of cones. Since these two
triangle are similar, we can write the
equations:
PQ/PR=PS/PT; PS=PF, PT=PF',
PQ/PR=PF/PF'; or PF/PQ=PF'/PR. Consider
P moving downward on upper branch of the
hyperbola. When P and F and F' become
collinear we have: PF=c-a, PQ=a-x,
PF'=2c-(c-a)=c+a, PR=2a-(a-x)=a+x and the
equation for x is: (c-a)/(a-x)=(a+c)/(a+x)-----(1)
x=a2
/c, where x is measured from the center
of FF'.

7. 7
Hyperbola eccentricity as a function
of the radii of Dandelin's Spheres
● If we note with r1
and r2
, r2
>r1
, the radii of the
Dandelin's spheres and d the distance between
their centers we can easily calculate FF'
(external tangent see slide #4) and QR (internal
tangent see slide #5) as:
● FF'=SQRT(d2
-(r2
-r1
)2)
=2c----------------(2)
● QR=SQRT(d2
-(r2
+r1
)2
)=2a--------------(3)
● And the eccentricity e=c/a>1 as:
● e=SQRT((d2
-(r2
-r1
)2
)/(d2
-(r2
+r1
)2
))-------------(4)

8. 8
Types of sections in a double cone
that can be treated with the model
described in here for ellipse
Circle-------Ellipse-----Hyperbola--Parabola
http://math2.org/math/algebra/conics.htmhttp://math2.org/math/algebra/conics.htm

9. 9
Conclusion
● Using the learning shifts suggested in a
previous posting, link below,
the Internet is found very helpful on fully
describing the hyperbola utilizing the synthetic
geometry in space and plane.
● The method shown could be applied easily to
other conics such as parabola.
https://www.linkedin.com/pulse/learning-shifts-internet-era-danut-dragoi?trk=mp-reader-card

10. 9
Conclusion
● Using the learning shifts suggested in a
previous posting, link below,
the Internet is found very helpful on fully
describing the hyperbola utilizing the synthetic
geometry in space and plane.
● The method shown could be applied easily to
other conics such as parabola.
https://www.linkedin.com/pulse/learning-shifts-internet-era-danut-dragoi?trk=mp-reader-card