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Quadratic surfaces standard form help

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  1. Jan 31, 2017 #1
    1. The problem statement, all variables and given/known data

    Suppose a quadratic equation in 3 variables is put into a standard form represents a hyperboloid of one sheet. This hyperboloid has the property that:

    • the cross section through z= 0 is a circle of radius 1;

    • the cross section through x= 1 is the two straight lines given (in the plane x = 1) by y = ± (1/2) z

    What is the standard form and the original equation in quadratic form?


    2. Relevant equations

    hyperboloid of one sheet is given by

    x^2/a^2+y^2/b^2-z^2/c^2=1


    3. The attempt at a solution

    using z=0

    x^2/a^2+y^2/b^2=1 which is an ellipse

    Given that it is a circle it means that b=a which means that it equals 1 as that is what the radius is.

    x^2+y^2-z^2/c^2=1

    How would I find C from here
     
    Last edited: Jan 31, 2017
  2. jcsd
  3. Jan 31, 2017 #2

    haruspex

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    Use the other piece of information:
     
  4. Jan 31, 2017 #3
    1+(1/4)z^2-z^2/c^2=1

    (1/4)z^2-z^2/c^2=0

    z^2((1/4)-c^2)=0

    (1/4)-c^2)=0

    c^2=1/4

    c=1/2

    therefore

    x^2+y^2-z^2/(1/2)^2=1
     
  5. Jan 31, 2017 #4

    pasmith

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    Why not simplify 1/(1/2)^2 as 4?
     
  6. Jan 31, 2017 #5

    haruspex

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    Check that step.
     
  7. Jan 31, 2017 #6
    that would mean that c=2
     
  8. Jan 31, 2017 #7

    haruspex

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    Yes.
     
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