Question

In what ratio does the $x$-axis divide the line segment joining the points $\left(-4,-6\right)$ and $\left(-1,7\right)$?

Find the coordinates of the point of division.

Answer 1

Compute the required ratio.

In the question, it is given that the $x$-axis divides the line segment joining the points $\left(-4,-6\right)$ and $\left(-1,7\right)$

We know that if a point $(x,y)$ divides the line segment joining the points $(x_1,y_1)$ and $({\mathrm{x}}_2,{\mathrm{y}}_2)$ in $m:n$. Then $\left(\mathrm{x},\mathrm{y}\right)=\left(\frac{{\mathrm{mx}}_2+{\mathrm{nx}}_1}{\mathrm{m}+\mathrm{n}},\frac{{\mathrm{my}}_2+{\mathrm{ny}}_1}{\mathrm{m}+\mathrm{n}}\right)$

Any point on the $x$-axis will be of type $\mathrm{y}=0$, so let the point be $\left(\mathrm{x},0\right)$.

Let the point divide the Line segment in ratio $k:1$.

By section formula,

$$\begin{gathered} 0=\frac{7\mathrm{k}-6}{\mathrm{k}+1} \\ \Rightarrow \mathrm{k}=\frac{6}{7} \end{gathered}$$

Therefore, The $x$-axis divides the line segment joining the points $\left(-4,-6\right)$ and $\left(-1,7\right)$ is $6:7$.

$$\begin{gathered} \mathrm{x}=\frac{-\mathrm{k}-4}{\mathrm{k}+1}\left[\because \mathrm{k}=\frac{6}{7}\right] \\ \Rightarrow \mathrm{x}=-\frac{{\displaystyle \raisebox{1ex}{$34$}\!\left/ \!\raisebox{-1ex}{$7$}\right.}}{{\displaystyle \raisebox{1ex}{$13$}\!\left/ \!\raisebox{-1ex}{$7$}\right.}} \\ \Rightarrow \mathrm{x}=\frac{-34}{13} \end{gathered}$$

Therefore, the coordinates of the point of division is $\left(\frac{-34}{13},0\right)$.

Hence the point $\left(\frac{-34}{13},0\right)$ divides the line segment in ratio $\frac{6}{7}$.