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

**Jenny20** · January 15th 2007, 04:27 PM

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.

**Soroban** · January 15th 2007, 05:08 PM

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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