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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
Center Radius r
(
)
,
k
9
Example 2
Center ? Radius ?
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.
Center ? Radius ?
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
  • Weve talked about this before
  • 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
    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.

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
have a minus sign in the middle
(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)
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