Elipse to a spheroid?

Question

I'm having a problem trying to prove that an elipse, given by $$ \xi = \left\{(x,y,z)\in \mathbb{R}^3 : y = 0, \frac{x^2}{a^2}+\frac{z^2}{c^2} = 1\right\} $$ if we rotate to the z-axis is a spheroid given by $$M = \left\{(x,y,z)\in \mathbb{R}^3 :\frac{x^2}{a^2} + \frac{y^2}{a^2}+\frac{z^2}{c^2} = 1 \right\}$$

Answer 1

You can rename the initial $x$-axis as $r$. (Because now it is something like the (perpendicular) distance from the $z$-axis) Then you can use polar coordinates for the $xy$-plane.

$$ x = r \cos \theta \\ y = r \sin \theta$$

and perhaps a trigonometric identity to get the final answer that you seek.