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

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

Standard Equation
With 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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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.

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• 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.

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• 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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• 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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• 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.

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• 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.

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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
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Example 3
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Center (-4 , 5) a 5 b 2
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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.

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• 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.

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• 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.

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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

• 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)
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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)