What 3D graph does $x^2+2z^2=1$ give?

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I am missing the 3D graph for the equation $x^2+2z^2=1$.

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3 Answers

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It's an ellipse in the $(x,z)$ plane which intersects $(\pm 1,0,0)$ and $(0,0,\pm\frac{1}{\sqrt{2}})$.

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$x^2+2z^2=1$ forms an ellipse in the $x$-$z$ plane. $y$ can be anything you like, so in three dimensions this looks like a cylinder centered around the $y$-axis except with an ellipse instead of a circle (the ellipse has a semi major axis of length $1$ in the $x$-direction and a semi minor axis of length $\frac{1}{\sqrt{2}}$ in the $z$-direction).

This figure is called an elliptical cylinder.

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It is an elliptic cylinder orthogonal to the $xz$ plane.

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