Find the ratio in which the $\displaystyle x$-axis divides the line segment joining the points $\displaystyle (-6, 5)$ and $\displaystyle (-4, -1)$. Also, find the point of intersection.

\[\begin{array}{ccccc} (-6,5) & \mathrm{k} & \mathrm{P} & 1 & (-4, \\ \hline \mathrm{A} & & (x, 0) & & \mathrm{B} \end{array} \] Let coordinates of P be ( \(\displaystyle x, 0\) ) and P divides the line segement AB in the ratio k : $\displaystyle 1$ \[\begin{aligned} & \left(\frac{-4 \mathrm{k}-6}{\mathrm{k}+1}, \frac{-\mathrm{k}+5}{\mathrm{k}+1}\right)=(x, 0) \\ & \Rightarrow \frac{-\mathrm{k}+5}{\mathrm{k}+1}=0 \Rightarrow \mathrm{k}=5 \end{aligned} \] Hence the required ratio is $\displaystyle 5$: $\displaystyle 1$ ∴ Coordinates of P are \(\displaystyle \left(\frac{-4 \times 5-6}{5+1}, 0\right)=\left(-\frac{13}{3}, 0\right)\)