Write the equation $7z^2=8x^2+8y^2$ in spherical form

Question

How can i write the equation $7z^2=8x^2+8y^2$ in spherical form

What i try

Put $x=\rho\sin \phi \cos \theta$ and $y=\rho\cos\phi\cos\theta$ and $z=\rho\cos\phi$

So equation is

$$7\rho^2\cos^2\phi=8\rho^2\sin^2\phi$$

$$\tan^2\phi=\frac{7}{8}$$

I did not understand how can i write above equation in spherical form because there are no $\rho$ present

Please also explain me is my answer is correct or not. Thanks

Answer 1

Yes, your solution is correct. It is the equation of a conical surface with half angle $\phi$ from the $z$ axis.

Answer 2

When you write $7\rho^2\cos^2\phi=8\sin^2\phi$ you missed $\rho$ on right, though final result is correct.

As $(\phi, \theta, \rho)$ is usual 3 dim space, then you have plane $\phi$ = const. It is well known Conical surface in $(x,y,z)$ coordinates.