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Math Help - z^2 = x^3+2xy^2-x

  1. #1
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    z^2 = x^3+2xy^2-x

    For the surface defined by z^2 = x^3+2xy^2-x, describe as best you can the traces in planes parallel to the yz-plane.

    Please teach me how to do this question. Thank you very much.
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  2. #2
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    Hello, Jenny!

    For the surface defined by z^2 \:= \:x^3+2xy^2-x,
    describe as best you can the traces in planes parallel to the yz-plane.

    A plane parallel to the yz-plane has the equation: . x \,=\,k (constant).

    In the equation let x = k\!:\;\;z^2\:=\:k^3 + 2ky^2 - k

    . . So we have: . z^2\:=\:2ky^2 + k^3 - k

    2k is a constant; call it a.
    k^3 - k is a constant; call it b.

    The equation has the form: . z^2\:=\:ay^2 + b\quad\Rightarrow\quad z^2 - ay^2 \:=\:b


    The traces are hyperbolas.

    If b is positive, the hyperbola is "vertical": \begin{array}{cc}\cup \\ \cap\end{array}

    If b is negative, the hyperbola is "horizontal": \supset\;\subset

    If b = 0, we have a pair of intersecting lines: z \:=\:\pm\sqrt{a}\,y

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