INTRODUCTION TO CONIC SECTIONS .ppt

Conic sections are the curves obtained by slicing a cone with a plane at various angles and positions. These curves play a vital role in geometry, algebra, and many real-world applications such as satellite orbits, optical systems, and architecture.

1. Introduction to Conic Sections

2. A CONIC SECTION IS A CURVE FORMED BY THE
INTERSECTION OF _________________________
a plane and a double cone.

5. Circles

6. Circles
The set of all points that are the same
distance from the center.
222
)()( rkyhx Standard Equation:
CENTER: (h, k)
RADIUS: r (square root)
(h , k)
r

7. Example 1
81)8()2(
22
 yx
h
k r²
Center:
Radius: r
),(
9
k (),h 82

8. Example 2
1)1(
22
yx
Center ?
Radius ?

9. Ellipses
Salami is often cut obliquely to obtain elliptical slices, which are
larger.

10. Ellipses
Basically, an ellipse is a squished
circle
Standard Equation:
Center: (h,k)
a: major radius, length from center to edge of circle
b: minor radius, length from center to top/bottom of circle
* You must square root the denominator
(h , k)
a
b

11. History
Early Greek astronomers thought that the planets
moved in circular orbits about an unmoving earth,
since the circle is the simplest mathematical curve.
In the 17th century, Johannes Kepler eventually
discovered that each planet travels around the sun in
an elliptical orbit with the sun at one of its foci.

12. On a far smaller
scale, the electrons
of an atom move in
an approximately
elliptical orbit with
the nucleus at one
focus.
Science

13. Any cylinder sliced on an
angle will reveal an ellipse in
cross-section
(as seen in the Tycho Brahe
Planetarium in
Copenhagen).

14. The ellipse has an important property that is used in the
reflection of light and sound waves.
Any light or signal that starts at one focus will be reflected to
the other focus.
Properties of Ellipses

15. The principle is also
used in the
construction of
"whispering galleries"
such as in St. Paul's
Cathedral in London.
If a person whispers
near one focus, he
can be heard at the
other focus, although
he cannot be heard at
many places in
between.

16. Example 3
1
4
)5(
25
)4(
22



 yx
a²
b
2
This must
equal 1
Center: (-4 , 5)
a: 5
b: 2

17. Parabolas

18. Parabolas
)(4)(
2
kyphx 
Standard Equations:
)(4)(
2
hxpky 
p>0 Opens UP Opens RIGHT
p<0 Opens DOWN Opens LEFT
vertex
vertex

19. One of nature's best
approximations to
parabolas is the
path of a projectile.

20. This discovery by Galileo in the 17th century
made it possible for cannoneers to work out
the kind of path a cannonball would travel if
it were hurtled through the air at a specific
angle.

21. Parabolas exhibit unusual and
useful reflective properties.
If a light is placed at the focus of
a parabolic mirror, the light will
be reflected in rays parallel to its
axis.
In this way a straight beam of
light is formed.
It is for this reason that parabolic
surfaces are used for headlamp
reflectors.
The bulb is placed at the focus
for the high beam and in front of
the focus for the low beam.

22. The opposite principle is
used in the giant mirrors in
reflecting telescopes and in
antennas used to collect
light and radio waves from
outer space:
...the beam comes toward
the parabolic surface and is
brought into focus at the
focal point.

23. Example 4
)5()2(
12
1
2
 yx
What is the vertex?How does it open?(-2 , 5)
opens
down
Example 5
2
)2(1255  yx
What is the vertex?How does it open?(0 , 2)
opens
right

24. The Hyperbola
If a right circular cone is
intersected by a plane
perpendicular to its axis, part
of a hyperbola is formed.
Such an intersection can
occur in physical situations as
simple as sharpening a pencil
that has a polygonal cross
section or in the patterns
formed on a wall by a lamp
shade.

25. Hyperbolas

26. Hyperbolas
Looks like: two parabolas, back to back.
Standard Equations:
1
)()(
2
2
2
2




b
ky
a
hx
1
)()(
2
2
2
2




b
hx
a
ky
Opens UP and DOWNOpens LEFT and RIGHT
Center: (h , k)
(h , k)
(h , k)

27. Hyperbolas – Transverse
Axis

28. Hyperbolas - Application
A sonic boom shock wave
has the shape of a cone,
and it intersects the ground
in part of a hyperbola. It
hits every point on this
curve at the same time, so
that people in different
places along the curve on
the ground hear it at the
same time. Because the
airplane is moving forward,
the hyperbolic curve moves
forward and eventually the
boom can be heard by
everyone in its path.

29. Example 6
1
4
)5(
25
)4(
22



 yx
Center: (-4 , 5)
Opens: Left and right

30. Name the conic section and its
center or vertex.

31. 25
22
yx

32. 1
22
yx

33. 2
)2(
12
1
1  yx

34. 2
)2(
8
1
3  xy

35. 4)2(
22
 yx

36. 1
925
22

yx

37. 1
16
)2(
4
)1(
22



 yx

38. 49)1()2(
22
 yx

39. 1)7()5(
22
 yx

40. xy6
2


41. 1
4
)1(
2
2


x
y

42. 1
5
)5(
17
)4(
22



 xy

43. Acknowledgements
http://hotmath.com/hotmath_help/topics/
parabolas.html
http://upload.wikimedia.org/wikipedia/
commons/8/85/Hyperbola_(PSF).png
http://www.funwearsports.com/NHL/CAPITALS/
WCDomedHockeyPuck.gif
Mathwarehouse.com
http://britton.disted.camosun.bc.ca/jbconics.htm
schools.paulding.k12.ga.us/.../
Introduction_to_Conics.ppt