STATEMENT_I: If two circles $x^2+y^2+2gx+2fy=0$ and  $x^2+y^2+2g^{1}x+2f^{1}y=0$ touch each other then $f^{1}g=f{g}^1$.

STATEMENT_II: Two circles touch each other if the line joining their centres is perpendicular to all possible common tangents.

Statement-I:

$c,o,c^{|}$ are collinear

$\begin{array}{l}\Rightarrow slopeof\overrightarrow{oc}=slopeof\overrightarrow{o{c}^{|}}\\ \Rightarrow \frac{f}{g}=-\frac{f^{|}}{g^{|}}\\ \Rightarrow f{g}^{|}=g{f}^{|}\end{array}$

Statement-II:

Two circles touch each other line joining of centres is not perpendicular in case of direct common tangents $l\text{\hspace{0.17em}and\hspace{0.17em}}{l}^1$

In case of transverse common tangent given statement is correct.