Answer
This is the equation of a sphere:
${x^2} + {y^2} + {\left( {z - 1} \right)^2} = 1$
The radius of the sphere is $1$ and its center is on the $z$-axis, at $\left( {0,0,1} \right)$.
Work Step by Step
We have $\rho = 2\cos \phi $.
Squaring both sides we get ${\rho ^2} = 4{\cos ^2}\phi $.
In spherical coordinates we have ${\rho ^2} = {x^2} + {y^2} + {z^2}$ and $z = \rho \cos \phi $. So, $\cos \phi = \frac{z}{\rho }$.
${\cos ^2}\phi = \frac{{{z^2}}}{{{\rho ^2}}} = \frac{{{z^2}}}{{{x^2} + {y^2} + {z^2}}}$
Thus,
${\rho ^2} = 4{\cos ^2}\phi $
${x^2} + {y^2} + {z^2} = \frac{{4{z^2}}}{{{x^2} + {y^2} + {z^2}}}$
${\left( {{x^2} + {y^2} + {z^2}} \right)^2} = 4{z^2}$
${x^2} + {y^2} + {z^2} = 2z$
${x^2} + {y^2} + {z^2} - 2z = 0$
${x^2} + {y^2} + {\left( {z - 1} \right)^2} - 1 = 0$
${x^2} + {y^2} + {\left( {z - 1} \right)^2} = 1$
This is the equation of a sphere with its center on the $z$-axis. The radius of the sphere is $1$ and its center is at $\left( {0,0,1} \right)$.
