Haymanon Conics by HC120218211529

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									Conic Sections

 By: Danielle Hayman
  Mrs. Guest Per. 4
What are conic sections?
• Conic sections are curves that are formed by the
  intersection of a cone and a plane. The four very familiar
  conics are knows as the circle, the ellipse, the parabola
  and the Hyperbola. The circle is made when the plane
  and the cone intersect making a closed curve. If the
  plane is perpendicular to the axis of the cone the conic is
  called an ellipse. If the plane is parallel to the line of the
  cone the conic is called a parabola. Then if the
  intersection is an open cone this conic is called a
  hyperbola, this plane will intersect both halves of the
  cone (two separate curves).
This is called a Circle.

• You can see in the picture that the plane
 is perpendicular to the axis of the cone.
         Distance formula
          2    2     2
         X +Y=r
This is called an Ellipse.

• An Ellipse is the set of all points (x,y)

     2                  2
 X       2   + Y   2   =1
 a             b
This is called a parabola

• This is the intersection of a right circular
  conical surface and a plane.
   Horizontal y = 4(-2)X
   FOCUS (O,P)
This is called a hyperbola.

• The intersection between a conical surface
 and a plane which cuts through both
 halves of the cone. Which creates two
 separate curves.
   2          2

  X    2   + Y 2 =1
   a          b
Conic sections

• Shown together.
Another way to see conics, and you can also
try this at home with a Styrofoam cup.
Translated conics
• Point (h,k) would be the vertex that belongs to the parabola and it is considered the
    center of other conics.
•   Translated conics:
•                  2         2    2
•   Circle: (x - h) + (y – k) = r
•   Horizontal Axis 2                          Vertical axis       2
•   Parabola: (y – k) = 4p(x – h)                           (x – h) = 4p (y – k)
•                    2         2                                  2         2
•   Ellipse: (x – h)2 + (y – k)2                            (x – h)2 + (y – k)2
•              a          b                                   a          b
•                       2         2
• Hyperbola: (x – h)2 - (y – k)2                           (y – k)2 - (x – h)2
•             a           b                                 a          b

• Notice when the parabola is horizontal the y comes before the x and the other way
    around when it is vertical, this is how you tell if the parabola is horizontal or vertical.
• Parabolas, Circles, Ellipses and hyperbolas can all be formed by
    intersecting a plane and a double- napped cone that is why they are called
    conic sections.
•   There are equations in x and y that have graphs that are not considered
•              2   2

• For example x + y = 0 is just one point and instead of adding that
    equation if you would subtract it then it would only be two intersecting
    lines and not a conic.
•     2               2

•   Ax + Bxy + Cy + Dx + Ey + F = 0 these type of conics are
•                         2

•   determined by B – 4Ac
•                             2

•   Ellipse and Circle : B – 4AC<0 the graph is a circle if A= C
•             2

•   Parabola B - 4AC = 0
•                 2
•   Hyperbola B – 4AC>0
Graph the Hyperbola.
Find the vertices and the foci of the
How are conics used in the real world?
• To build things such as a sculpture at Fermi National
   Accelerator Laboratory. You can see all the “standard
Conics used in real life.

• The parabola is in the McDonalds sign.
• For this project I used
• www.kn.att.com.com/wired/fil/pages/listc
• www.mathworld.com
• http://math2.org/math/algebra/conics.htm
• www.google.com
• Our class Algebra two book

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