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Math Help - a little bit confused! conics

  1. #1
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    Unhappy a little bit confused! conics

    the question is as follows: the vertex angle of a double napped cone is 105 degrees the angle 0 is the angle measured from the axis to the cutting plane. for what angle 0, or range of values for 0, does the cutting plane need to interescet the conical surface in order to generate each of the following conics?
    a. Hyperbola
    b. ellipse
    C.circle
    d. parabola

    how would I go do this question cuz first off I don't get what it wants from me!
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  2. #2
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    Hello, George!

    If no one has ever explained Conic Section (and conic curves) to you,
    . . no wonder that you don't have a clue . . .

    The vertex angle of 105° is very inconsiderate.
    I'll change it to 120°.


    The vertex angle of a double napped cone is 105°.
    The angle \theta is the angle measured from the axis to the cutting plane.
    For what angle \theta, or range of values for \theta, does the cutting plane need to
    intersect the conical surface in order to generate each of the following conics?
    (a) hyperbola . . (b) Ellipse . . (c) circle . . (d) parabola

    If the plane is perpendicular to the axis of the cone (θ = 90°),
    the intersection is a circle.
    Code:
                  *
                 /:\
                / : \
               /  :  \
              /60°:   \
             /    :    \
          o o o o o o o o o
           /     θ:      \
          /       :       \

    If the cutting plane is parallel to the slant of the cone (θ = 30°)
    . . the intersection is a parabola.
    Code:
                  *
                 /:\
                / : \o
               /  : o\
              /   :o  \
             /    o    \
            /    o:     \
           /    o :      \
          /    o θ|       \

    In between those two angles (30° < θ < 90°)
    . . the intersection is an ellipse.
    Code:
                  *
                 /:\
                / : \
               /  :  \     o
              /   :   \ o
             /    :  o \
            /     o     \
           /   o  :      \
          / o   θ :       \
         o        :        \

    If the plane is steeper than that of the parabola (θ < 30°),
    . . the intersection is a hyperbola.
    Note that the plane cuts both nappes of the cone.
    Code:
          \       :     o /
           \      :      /
            \     :    o/
             \    :    /
              \   :   o
               \  :  /
                \ : /o
                 \:/
                  * o
                 /:\
                / :o\
               /  :  \
              /   o   \
             /    :    \
            /    o:     \
           /      :      \
          /     o :       \
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  3. #3
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    check!

    a.circle: (θ = 90&#176
    b. parabola: 26.25&#176;
    c. an ellipse: 26.25&#176;< θ< 90&#176;
    d. hyperbola: θ< 26.25&#176;

    are these correct???
    thanks!
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  4. #4
    Grand Panjandrum
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    Quote Originally Posted by george93 View Post
    a.circle: (θ = 90°)
    b. parabola: 26.25°
    105/2=52.5 degrees if we are still talking about the double cone with
    vertical angle of 105 degrees.
    c. an ellipse: 26.25°< θ< 90°
    d. hyperbola: θ< 26.25°
    Then these are:

    c. an ellipse 52.5°<θ<90°
    d. hyperbola: θ< 52.5°

    (Soroban appears to have divided the vertical angle by 2 twice rather than once
    in his example as far as I can tell)

    RonL
    Attached Thumbnails Attached Thumbnails a little bit confused! conics-gash.jpg  
    Last edited by CaptainBlack; October 31st 2006 at 11:25 AM.
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  5. #5
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    Red face hyperbola

    um.. thanks It makes sesns to me! okay I have one question!
    a degenerate form of the hyperbola is a pair of intersecting lines. how would I be able to postions an intersecting plane to intersect the cone described in the question above to obtain a degenerate hyperbola?

    I am confused on this question o..
    ps I got an assignemt that was all about the first question and I got 89% thanks this website rocks!!!
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  6. #6
    Grand Panjandrum
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    Quote Originally Posted by george93 View Post
    um.. thanks It makes sesns to me! okay I have one question!
    a degenerate form of the hyperbola is a pair of intersecting lines. how would I be able to postions an intersecting plane to intersect the cone described in the question above to obtain a degenerate hyperbola?

    I am confused on this question o..
    ps I got an assignemt that was all about the first question and I got 89% thanks this website rocks!!!
    Think about planes which pass through the vertex.

    RonL
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  7. #7
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    so...

    a degenerate form of the hyperbola is a pair of intersecting lines. how would I be able to postions an intersecting plane to intersect the cone described in the question above to obtain a degenerate hyperbola?
    so is this what the question wants:

    Thanks for your help!
    Attached Thumbnails Attached Thumbnails a little bit confused! conics-untitled.jpg  
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