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## Introduction to Conics

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Title: Introduction to Conics

1
Introduction to Conic Sections
2
• A conic section is a curve formed by the
intersection of _________________________

a plane and a double cone.
3
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4
History
• Conic sections is one of the oldest math subject
studied.
• The conics were discovered by Greek mathematician
Menaechmus (c. 375-325 BC)
• Menaechmuss intelligence was highly regarded he
tutored Alexander the Great.

5
History
• Appollonius (c. 262-190 BC) wrote about conics in
his series of books simply titled Conic
Sections.
• Appollonious nickname was the Great Geometer
• He was the first to base the theory of all three
conics on sections of one circular cone.
• He is also the one to give the name ellipse,
parabola, and hyperbola.

6
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7
Circles
• The set of all points that are the same distance
from the center.

Standard Equation
With CENTER (h, k) RADIUS r (square root)
8
Warm-Up
-h

-k
(
)
,
k
9
Example 2
10
Warm-Up
When the tardy bell rings Please have out your
homework, pen to check and pencil and be working
on this warm-up in your spiral below yesterday.
1. 2.
11
The Ellipse
• Tilt a glass of water and the surface of the
liquid acquires an elliptical outline.
• Salami is often cut obliquely to obtain
elliptical slices which are larger.

12
• -The 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.

13
• On a far smaller scale, the electrons of an atom
move in an approximately elliptical orbit with
the nucleus at one focus.

14
• Any cylinder sliced on an angle will reveal an
ellipse in cross-section
• (as seen in the Tycho Brahe Planetarium in
Copenhagen).

15
• 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 foci will
be reflected to the other foci.

Foci
Foci
16
• 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.

17
• Statuary Hall in the U.S. Capital building is
elliptic.
• It was in this room that John Quincy Adams, while
a member of the House of Representatives,
discovered this acoustical phenomenon.
• He situated his desk at a focal point of the
elliptical ceiling, easily eavesdropping on the
private conversations of other House members
located near the other focal point.

18
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19
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20
• The ability of the ellipse to rebound an object
starting from one focus to the other focus can be
demonstrated with an elliptical pool table.
• When a ball is placed at one focus and is thrust
with a cue stick, it will rebound to the other
focus.
• If the pool table is live enough, the ball will
continue passing through each focus and rebound
to the other.

21
Ellipse
• Basically an ellipse is a squished circle

Center (h , k) a major radius (horizontal),
length from center to edge of circle b minor
radius (vertical), length from center to
top/bottom of circle
You must square root the denominator
22
Example 3
2
Center (-4 , 5) a 5 b 2
23
Parabola
vertex
vertex
• a U-shaped graph

This equation opens left or right
This equation opens up or down
HOW DO YOU TELLLOOK FOR THE SQUARED VARIABLE
• Vertex (h , k)
• If there is a negative in front of the squared
variable, then it opens down or left.
• If there is NOT a negative, then it opens up or
right.

24
• One of nature's best known approximations to
parabolas is the path taken by a body projected
upward, as in the parabolic trajectory of a golf
ball.

25
• The easiest way to visualize the path of a
projectile is to observe a waterspout.
• Each molecule of water follows the same path and,
therefore, reveals a picture of the curve.

26
• 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.

27
• 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
• The bulb is placed at the focus for the high beam
and in front of the focus for the low beam.

28
• 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.

29
Example 4
opens down
What is the vertex? How does it open?
(-2 , 5)
opens right
What is the vertex? How does it open?
(0 , 2)
30
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.

31
Hyperbolas
• What I look liketwo parabolas, back to back.

This equation opens up and down
This equation opens left and right
Have I seen this before? Sort ofonly now we
(h , k)
Center (h , k)
32
Example 6
Center (-4 , 5) Opens Left and right
33
What am I?
Name the conic section and its center or vertex.
34
circle (0,0)
35
hyperbola (0,0)
36
parabola vertex (1,-2)
37
parabola vertex (-2,-3)
38
circle (2,0)
39
ellipse (0,0)
40
hyperbola (1,-2)
41
circle (-2,-1)
42
hyperbola (-5,7)
43
parabola vertex (0,0)
44
hyperbola (0,1)
45
ellipse (-5,4)