The ratio in which the line segment joining the points $\left(3,-4\right)and\left(-5,6\right)$  is divides by the x-axis, is

Let P is the point on x-axis which divides the line segment joining points in the ratio $\lambda :1,$  then , $x=\frac{-5\lambda +3}{\lambda +1},y=\frac{6\lambda -4}{\lambda +1}$

but as it is know for point on x-axis, then its y-coordinate$=0$

$\Rightarrow$     $\frac{6\lambda -4}{\lambda +1}=0\Rightarrow \lambda =\frac{2}{3}$