Determine the ratio in which the line $\displaystyle 3 \mathrm{x}+\mathrm{y}-9=0$ divides the line segment joining the points $\displaystyle (1,3)$ and $\displaystyle (2,5)$. Find the point of intersection.

Let the line \(\displaystyle 3 \mathrm{x}+\mathrm{y}-9=0\) divides the line segment joining the points A \(\displaystyle (1,3)\) and \(\displaystyle \mathrm{B}(2,5)\) in the ratio \(\displaystyle \mathrm{k}: 1\) at point \(\displaystyle \mathrm{P}(\mathrm{x}, \mathrm{y})\). Since point \(\displaystyle \mathrm{P}(\mathrm{x}, \mathrm{y})\) lies on line \(\displaystyle 3 \mathrm{x}+\mathrm{y}-9=0\), \[\begin{aligned} & \therefore 3\left(\frac{2 \mathrm{k}+1}{\mathrm{k}+1}\right)+\left(\frac{5 \mathrm{k}+3}{\mathrm{k}+1}\right)-9=0 \\ & \Rightarrow \mathrm{k}=\frac{3}{2} \end{aligned} \] Therefore, the required ratio is $\displaystyle 3$ : 2. Coordinates of P are \(\displaystyle \left(\frac{1 \times 2+2 \times 3}{2+3}, \frac{3 \times 2+5 \times 3}{2+3}\right)=\left(\frac{8}{5}, \frac{21}{5}\right)\)