Spherical Coordinates — Question 4

Question 4

Rewrite the sphere $x^2+y^2+z^2=6z$ in spherical coordinates and identify its geometry.

Tasks

1. Obtain the nontrivial spherical equation.

2. Find its Cartesian center and radius.

3. Explain the role of $\rho=0$ when simplifying.

Question 4 – Solution

Strategy. Substitute $x^2+y^2+z^2=\rho^2$ and $z=\rho\cos\phi$, while preserving any solution lost by division.

Step 1: Convert $$\rho^2=6\rho\cos\phi\quad\Longleftrightarrow\quad \rho(\rho-6\cos\phi)=0.$$ The usual surface equation is $$\boxed{\rho=6\cos\phi},\qquad 0\le\phi\le\frac\pi 2,$$ and it includes the origin at $\phi=\pi/2$.

Step 2: Identify Completing the square in Cartesian form gives $$x^2+y^2+(z-3)^2=9.$$ It is the sphere centered at $\boxed{(0,0,3)}$ with radius $\boxed{3}$.

Step 3: Origin check Division by $\rho$ would temporarily discard $\rho=0$. The origin satisfies the original equation and is recovered by $\rho=6\cos(\pi/2)=0$; thus no point is absent from the final parametrized surface.