In what ratio does the $ x $-axis divide the line segment joining the points $ (-4,-6) $ and $ (-1,7) $ ? Find the coordinates of the point of division.

Given:

The line segment joining the points \( (-4,-6) \) and \( (-1,7) \) 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 $A(-4, -6)$ and $B(-1, 7)$ 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,

$P(x, 0)=(\frac{m \times (-1)+n \times (-4)}{m+n}, \frac{m \times 7+n \times(-6)}{(m+n)})$

$\Rightarrow \frac{7m-6n}{m+n}=0$

$\Rightarrow 7m-6n=0$

$\Rightarrow 7m=6n$

$\Rightarrow \frac{m}{n}=\frac{6}{7}$

$\Rightarrow m:n=6:7$

This implies,

$x=\frac{6(-1)+7(-4)}{6+7}$

$=\frac{-6-28}{13}$

$=\frac{-34}{13}$ 

The ratio of the division is $6:7$ and the coordinates of the point of division are $(\frac{-34}{13},0)$.