If the segment of the line between the lines $x–y+2=0$ and $x+y+4=0$ is bisected at the origin, equation of the line is 

Let the required line meet the given lines at $\left({\alpha }_1,{\beta }_1\right)$ and $\left({\alpha }_2,{\beta }_2\right)$ respectively.
then ${\alpha }_1-{\beta }_1,+2=0, {\alpha }_2+{\beta }_2+4=0$
Also $\frac{{\alpha }_1+{\alpha }_2}{2}=0, \frac{{\beta }_1+{\beta }_2}{2}=0$
$\Rightarrow {\alpha }_2=-1,{\beta }_2=-3$ and ${\alpha }_1=1,{\beta }_1=3$
Equation of the line passing through $(1, 3)$ and $(0, 0)$ is
           $y=3x \Rightarrow y-3x=0$