10'3 Conics The Parabola

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10.3 ConicsThe Parabola
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Definition of a Parabola

• A Parabola is the set of all points in a plane
  that are equidistant from a fixed line (the
  directrix) and a fixed point (the focus) that is
  not on the line.

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Animation of Definition
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Applications of a Parabola

• Some comets shoot through our solar system along
  parabolic paths with the Sun at the focus. Notice
  how the comet speeds up as it gets closer to the
  Sun... It's a gravity thing!
• Look at the comet's tail. The solar wind always
  blows the tail away from the Sun. By the way, I
  highly recommend that you take an astronomy class
  in college. You'll love it!)

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Parabolic applications
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Parabolic applications

• Automobile Headlights
•         An automobile headlight is another
  example of a Paraboloid of Revolution --taking a
  parabola and rotating it about its axis of
  symmetry.        The smooth inner surface of the
  headlight is a glass reflector upon which bright
  aluminum has been deposited. This part is a
  powerful reflector.
•         A parabolic reflector has the property
  that if a light source is placed at the focus of
  the reflector, the light rays will reflect from
  the mirror as rays parallel to the axis.       
  This is used in auto headlights to give an
  intense concentrated beam of light.

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Parabolic applications

• The principle of the parabolic reflector may have
  been discovered in the 3rd century BC by the
  geometer Archimedes, who, according to a legend
  of debatable veracity constructed parabolic
  mirrors to defend Syracuse against the Roman
  fleet, by concentrating the sun's rays to set
  fire to the decks of the Roman ships.

Le Four Solaire at Font-Romeur There is a
reflector in the Pyrenees Mountains that is 8
stories high. It cost two million dollars to
build and it took ten years to build it. It is
made of 9,000 mirrors arranged in a parabolic
mirror. It can reach 6,000 degrees Fahrenheit
just from the Sun!
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Parabolic applications
Gallileo was the first to show that the path an
object thrown in space is a parabola.
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Parabolic applications
The cables that act as suspension are parabolas.
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Standard Forms of the Parabola

• The standard form of the equation of a parabola
  with vertex at the origin is
• y2 4px or x2 4py.
• The graph illustrates that for the equation on
  the left, the focus is on the x-axis, which is
  the axis of symmetry. For the equation of the
  right, the focus is on the y-axis, which is the
  axis of symmetry.

y
x
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Example 1

• Find the focus and directrix of the parabola
  given by

4p 16p 4Focus (0,4) and directrix y -4
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Example 1
Find the focus and directrix of the parabola
given by x2 -8y. Then graph the parabola.
The given equation is in the standard form x2
4py, so 4p -8. x2 -8y
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Example 1 cont'd.
Find the focus and directrix of the parabola
given by x2 -8y. The graph the parabola.
Because p -2, p lt 0, the parabola opens
downward. Using this value for p, we
obtain Focus (0, p) (0, -2) Directrix
y - p y 2.
To graph x2 -8y, we use test points. If we
assign y a value that makes the right side a
perfect square, it makes it easier. If y -2,
then x2 -8(-2) 16, so x is 4 and 4. The
parabola passes through the points (4, -2) and
(-4, -2).
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Example 2
Find the standard form of the equation of a
parabola with focus (5, 0) and directrix x -5.
The focus is (5, 0). implies the focus is on the
x-axis. Use the standard form of the equation
in which x is not squared, y2 4px. We need to
determine the value of p. The focus, located at
(p, 0), is p units from the vertex, (0, 0). then
p 5. We substitute 5 for p into y2 4px y2 4
5x or y2 20x.
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Standard form if vertex is not at the origin
(y k)2 4p(x h). (x h)2 4p(y
k).
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Example 3
Find the vertex, focus, and directrix of the
parabola given by y2 2y 12x 23 0. Then
graph the parabola.
Vertex ( 2 , -1 )
(y 1)2 -12x 24
p -3
(y 1)2 -12(x 2)
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Example 3 cont'd.
(y 1)2 -12(x 2)
Vertex ( 2 , -1 )
y is the squared term and p is negative -3
Directrix is p units from the vertex x 5
Focus pt. is p units from the vertex (-1, -1)
Sample point Let x -1 (y
1)2 36 (y 1) 6 y - 1
6 (-1, 5) and ( -1 -7)

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The Latus Rectum and Graphing Parabolas

• The latus rectum of a parabola is a line segment
  that passes through its focus, is parallel to its
  directrix, and has its endpoints on the parabola.
• The length of a parabola's latus rectum is 4p,
  where p is the distance from the focus to the
  vertex.

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Test tomorrowCh 9 - 10.3