Find the ratio in which line $\displaystyle \mathrm{y}=\mathrm{x}$ divides the line segment joining the points $\displaystyle (6, -3)$ and $\displaystyle (1, 6)$.

Let the ratio be \(\displaystyle \boldsymbol{k}\) : $\displaystyle 1$ \[\begin{aligned} & x=\frac{k+6}{k+1} \\ & y=\frac{6 k-3}{k+1} \end{aligned} \] \(\displaystyle \mathbf{P}(\mathbf{x}, \mathbf{y})\) lies on \(\displaystyle \mathbf{y}=\mathbf{x}\) \[\begin{aligned} & \Rightarrow \mathrm{k}+6=6 \mathrm{k}-3 \\ & \Rightarrow \mathrm{k}=\frac{9}{5} \end{aligned} \] Ratio is $\displaystyle 9$ : $\displaystyle 5$