Graphing $g(x,y,z): Circle vs Ellipse?

[NoName3]

I'm told that graph of the surface $g(x,y,z) = x^2+y^2+4z^2 = 1$ looks like:

Is that correct? And if so, I've the following question. When considering the slices the graph of $g(x,y, z)$ is a circle in the $xy$ plane and an ellipse in the $xz$ plane and $zy$ plane. The circle is bigger than the ellipses:

So what happened to the circle when graphing $g(x, y, z)$? How come the whole thing becomes an ellipsoid?

 

[jvicens]

Hello,

Yes, the graph of the surface $g(x,y,z) = x^2+y^2+4z^2 = 1$ does indeed look like an ellipsoid. This is because the equation represents a three-dimensional shape that is symmetrical in all three dimensions. The slices of the graph in the $xy$, $xz$, and $yz$ planes are circles and ellipses because they are two-dimensional representations of the three-dimensional shape.

When graphing the equation $g(x,y,z)$, the circle in the $xy$ plane becomes an ellipse because the $z$ variable is also included in the equation. This means that as we move along the $z$ axis, the value of $g(x,y,z)$ changes and the resulting graph is no longer a perfect circle. The whole thing becomes an ellipsoid because the $z$ variable affects the overall shape of the graph in all three dimensions.

I hope this helps to clarify any confusion. Please let me know if you have any further questions.