Answer
$VII$
Work Step by Step
The equation $x^{2}+4y^{2}+9z^{2}=1$ is an ellipsoid, so it can be either Figure $IV$ or Figure $VII$.
If we write the given equation in the standard form of an ellipsoid centered at the origin, we will have
$\frac{x^{2}}{1^{2}}+\frac{4y^{2}}{(1/2)^{2}}+\frac{z^{2}}{(1/3)^{2}}=1$
The numbers $1,1/2$ and $1/3$ are the number of units you go along either side of the $x-, y-$ and $z$-axes, respectively, from the origin, that is, the center of the ellipsoid, to get to the surface of the ellipsoid.
Since we can visualize that Figure $VII$, has the ellipsoid more elongated along the x-axis and that 1 (the number under $x^{2})$ is greater than $1/2$ and $1/3$, it must be Figure $VII$.
