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Thread: 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 $\displaystyle z^2 \:= \:x^3+2xy^2-x$,
    describe as best you can the traces in planes parallel to the yz-plane.

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

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

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

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

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


    The traces are hyperbolas.

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

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

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

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