Find the ratio in which the line segment joining $A(1, -5)$ and $B(-4, 5)$ is divided by the $x$-axis. Also,find the coordinates of the point of division.

(A) Let the ratio in which the line segment joining $A(1, -5)$ and $B(-4, 5)$ is divided by the $x$-axis be $k: 1$.
Using the section formula,the coordinates of the point of division are $\left(\frac{-4k + 1}{k + 1}, \frac{5k - 5}{k + 1}\right)$.
Since the point lies on the $x$-axis,its $y$-coordinate must be $0$.
Therefore,$\frac{5k - 5}{k + 1} = 0$.
This implies $5k - 5 = 0$,so $k = 1$.
The ratio is $1: 1$.
Substituting $k = 1$ into the $x$-coordinate expression: $\frac{-4(1) + 1}{1 + 1} = \frac{-3}{2}$.
Thus,the point of division is $\left(\frac{-3}{2}, 0\right)$.