Could someone help me with this problem:
The change in spherical coordinates is given by the equations:
$x= \rho \sin \phi \cos \theta \tag{I}$ $y= \rho \sin \phi \sin \theta \tag{II}$ $z = \rho \cos \phi \tag{III}$
The set of points $A = \{(x, y, z)\in\mathbb{R^3} \mid \rho=\text{constant}\}$ is a spherical surface.
The set of points $A = \{(x, y, z)\in \mathbb{R^3} \mid \theta =\text{constant}\}$ is a vertical semiplane passing through the $Oz$ axis.
And the set of points $A = \{(x, y, z)\in \mathbb{R^3} \mid \phi=\text{constant}\}$ is a cone.
Based on this theme, mark the alternative that correctly indicates the equation, in spherical coordinates, that describes the sphere:
$x^2 + y^2 + (z - a) ^2 = b^2 \tag{IV}$
Resolution:
I replaced (I), (II) and (III) in (IV) and arrived in:
$$\rho ^2 + a^2 -2a \rho \cos \phi= b^2 \text{ or } \rho^2 = b^2 -a^2 +2a \rho \cos \phi.$$
But the feedback says that the answer must be $\rho = b^2 -a^2 +2a \rho \cos \phi$
I don't understand where I'm going wrong.