Introduction to Conics

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

a plane and a double cone.
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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.

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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.

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Circles
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Circles

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

Standard Equation
CENTER (h, k) RADIUS r (square root)
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Example 1
h
r²
k
Center Radius r
)
(
,
k
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Example 2
Center ? Radius ?
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Ellipses

• Salami is often cut obliquely to obtain
  elliptical slices, which are larger.

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Ellipses

• Basically, an ellipse is a squished circle

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
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• 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.

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Science

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

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

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Properties of Ellipses

• 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.

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• 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.

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Example 3
Center (-4 , 5) a 5 b 2
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Parabolas
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Parabolas
vertex
vertex
pgt0 Opens UP Opens RIGHT plt0 Opens
DOWN Opens LEFT
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• One of nature's best approximations to parabolas
  is the path of a projectile.

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• 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.

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• 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.

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• 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.

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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)
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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.

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Hyperbolas
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Hyperbolas

• Looks like two parabolas, back to back.

Center (h , k)
Opens UP and DOWN
Opens LEFT and RIGHT
(h , k)
(h , k)
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Hyperbolas Transverse Axis
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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.
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Example 6
Center (-4 , 5) Opens Left and right
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What am I?
Name the conic section and its center or vertex.
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circle (0,0)
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hyperbola (0,0)
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parabola vertex (1,-2)
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parabola vertex (-2,-3)
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circle (2,0)
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ellipse (0,0)
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hyperbola (1,-2)
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circle (-2,-1)
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hyperbola (-5,7)
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parabola vertex (0,0)
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hyperbola (0,1)
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ellipse (-5,4)
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• Acknowledgements
• http//hotmath.com/hotmath_help/topics/parabolas.h
  tml
• http//upload.wikimedia.org/wikipedia/commons/8/85
  /Hyperbola_(PSF).png
• http//www.funwearsports.com/NHL/CAPITALS/WCDomedH
  ockeyPuck.gif
• Mathwarehouse.com
• http//britton.disted.camosun.bc.ca/jbconics.htm
• schools.paulding.k12.ga.us/.../Introduction_to_Con
  ics.ppt