Question

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.

Answer 1

Using the section formula, if a point $(x,y)$ divides the line joining the points $(x_1,y_1)$ and $(x_2,y_2)$ in the ratio $m:n$, then

$(x,y)=(\frac{m{x}_2+n{x}_1}{m+n},\frac{m{y}_2+n{y}_1}{m+n})$

We know if a point is on x-axis, then y coordinate is zero.

So, $(x,0)=(\frac{m{x}_2+n{x}_1}{m+n},\frac{m{y}_2+n{y}_1}{m+n})$

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

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

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

$\Rightarrow 7m-6n=0$

$\Rightarrow 7m=6n$

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

$x=\frac{m{x}_2+n{y}_1}{m+n}$

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

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

$\Rightarrow x=\frac{-34}{13}$

$\therefore$ co-ordinates $[\frac{-34}{13},0]$.