I is true. Note that the centre of $C_1$ is
$\begin{array}{rcl}
G_1 & = & \left( -\dfrac{8}{2}, – \dfrac{-4}{2} \right) \\
G_1 & = & (-4, 2)
\end{array}$
Rewrite the equation of $C_2$ to general form, we have $x^2+y^2+4x-2y-\dfrac{5}{2} =0$. Then the centre of $C_2$ is
$\begin{array}{rcl}
G_2 & = & \left(- \dfrac{4}{2}, -\dfrac{-2}{2} \right) \\
G_2 & = & (-2,1)
\end{array}$
Consider the slope of $G_1O$, we have
$\begin{array}{rcl}
m_{G_1O} & = & \dfrac{2-0}{-4-0} \\
m_{G_1O} & = & \dfrac{-1}{2}
\end{array}$
Consider the slope of $G_2O$, we have
$\begin{array}{rcl}
m_{G_2O} & = & \dfrac{1-0}{-2-0} \\
m_{G_2O} & = & \dfrac{-1}{2}
\end{array}$
Since $m_{G_1O} = m_{G_2O} = \dfrac{-1}{2}$ and $O$ is the common point of $G_1O$ and $G_2O$, then $G_1$, $G_2$ and $O$ are collinear.
II is not true. The radius of $C_1$
$\begin{array}{cl}
= & \sqrt{(-4)^2 + (2)^2 -(-5)} \\
= & 25
\end{array}$
The radius of $C_2$
$\begin{array}{cl}
= & \sqrt{(-2)^2 + (1)^2 -(\dfrac{-5}{2})} \\
= & \sqrt{\dfrac{15}{2}}
\end{array}$
Therefore, the radii of $C_1$ and $C_2$ are not equal.
III is not true. The distance between $G_1$ and $O$
$\begin{array}{cl}
= & \sqrt{(-4-0)^2+(2-0)^2} \\
= & \sqrt{20}
\end{array}$
The distance between $G_2$ and $O$
$\begin{array}{cl}
= & \sqrt{(-2-0)^2 + (1-0)^2 } \\
= & \sqrt{5}
\end{array}$
Therefore, $O$ is no equidistant from $G_1$ and $G_2$.
