The slope of a line is $3$ times the slope of the other line and the tangent of the angle between them is $4/13$, sum of their slopes is equal to

Let the slopes of the given lines be $m$ and $3$ m. if $\theta$ is the angle between them then

$\begin{array}{l} \tan \theta =\frac{|3m-m|}{1+\left(3m\right)\left(m\right)}=\frac{4}{13}\\ \Rightarrow \pm 2m\times 13=4\left(1+3m^2\right)\\ \Rightarrow 6m^2\pm 13m+2=0\end{array}$
$\Rightarrow m=\frac{1}{6}$ or $2$ and $m=-\frac{1}{6}$ or $-2.$
For $m=2,$ the sum of the slopes is $4m=8$