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# 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.

Given:

The line segment joining the points $A(1, -5)$ and $B(-4, 5)$ is divided by the x-axis.

To do:

We have to find the ratio of division and coordinates of the point of division.

Solution:

The point which divides the given line segment lies on x-axis.

This implies,

Its ordinate is $0$.

Let the point $P(x, 0)$ intersects the line segment joining the points $A(1, -5)$ and $B(-4, 5)$ in the ratio $m : n$.

Using section formula, we have,

$(x, y)=(\frac{mx_{2}+nx_{1}}{m+n}, \frac{my_{2}+ny_{1}}{m+n})$

Therefore,

$P(x, 0)=(\frac{m \times (-4)+n \times 1}{m+n}, \frac{m \times 5+n \times(-5)}{(m+n)})$

$\Rightarrow \frac{5m-5n}{m+n}=0$

$\Rightarrow 5m-5n=0$

$\Rightarrow 5m=5n$

$\Rightarrow \frac{m}{n}=\frac{5}{5}$

$\Rightarrow m:n=1:1$

This implies,

$x=\frac{1(-4)+1(1)}{1+1}$

$=\frac{-4+1}{2}$

$=\frac{-3}{2}$

The ratio of the division is $1:1$ and the coordinates of the point of division are $(\frac{-3}{2},0)$.

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