Find an equation in spherical coordinates for the surface represented by the rectangular equation

Question

The rectangular equation is $$x^2+y^2-8z^2=0$$ $$x^2+y^2=8z^2$$ Know in the relationship between rectangular and spherical coords. we can manipulate our given to fit the form: $$x^2+y^2+z^2=9z^2$$ $$\rho^2=x^2+y^2+z^2, \space z=\rho\cos(\phi)$$ $$\rho^2=9\rho^2\cos^2(\phi)$$ $$1=9\cos^2(\phi)$$ $$\frac{1}{3}=\cos(\phi)$$ $$\arccos\left(\frac{1}{3}\right)=1.23 \space rads$$ And so the equation in spherical coords. is $\phi=1.23$ I know my math is correct but I have the wrong answer so ilm not sure where I went wrong.

Answer 1

I think the best way to do this would be the following: $$8z^2=x^2+y^2$$ $$z^2=\frac{x^2+y^2}{8}$$ now convert to cylindrical coordinates: $$z^2=\frac{r^2}{8}$$ to do this we know: $$\rho^2=r^2+z^2,\,\theta=\theta,\,\cos(\varphi)=\frac{z}{\sqrt{r^2+z^2}}$$ now try and manipulate our equation