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conics ppt

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.
Standard Equation:
222
)()( rkyhx 
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. •Salami is often cut obliquely to obtain
elliptical slices, which are larger.
Ellipses

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. •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.
The Hyperbola

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