Spherical Coordinates — Question 3

Question 3 – Solution

Strategy Multiply by $\rho$ and use $\rho^2=x^2+y^2+z^2$ and $\rho\cos\phi=z$.

Conversion $$\rho^2=4\rho\cos\phi \quad\Longrightarrow\quad x^2+y^2+z^2=4z.$$ Completing the square, $$\boxed{x^2+y^2+(z-2)^2=4}.$$

Geometry The surface is a sphere of radius $2$ centered at $(0,0,2)$. It contains the origin because the distance from the origin to the center is exactly the radius. In the original spherical equation, $\rho=0$ and $\phi=\pi/2$ represent the origin and satisfy $0=4\cos(\pi/2)$.