Converting to spherical equation

Question

If i have the function 2$z^2$ = $x^2 + y^2$, how would i convert that to spherical?

I know $x^2 + y^2$ = $r^2$, but then I get the equation 2 = $r^2$/$z^2$ and I am stuck there.

Any suggestions?

Answer 1

Hint:

You are thinking of $2D$ polar co-ordinates. In $3D$ spherical co-ordinates, $$\begin{align}x&=r\cos\phi\sin\theta\\y&=r\sin\phi\sin\theta\\z&=r\cos\theta\end{align}$$ Can you do it from here?

Answer 2

Spherical coordinates have $x^2+y^2+z^2 = r^2$, which is the definition of a sphere centered at the origin, and $z = r\cos\theta$. You also have the equation

$$ x^2+y^2+z^2 = 3z^2$$

by simply adding $z^2$ to both sides.