Given, equation of line $y-x+2=0$ and co-ordinates of points $\left(x_1, y_1\right)=(3,-1)$ and $\left(x_2, y_2\right)=(8,9)$. We know that if the ratio in which a line $a x+b y+c=0$ is divided by points $\left(x_1, y_1\right)$ and $\left(x_2, y_2\right)$ is $\lambda: 1$, then intersecting point $\left(\frac{\lambda x_2+x_1}{\lambda+1}, \frac{\lambda y_2+y_1}{\lambda+1}\right)$ lies on $a x+b y+c=0$. Thus any point on the line joining $(3,-1)$ and $(8,9)$ dividing it in the ratio $\lambda: 1$ is $\left(\frac{8 \lambda+3}{\lambda+1}, \frac{9 \lambda-1}{\lambda+1}\right)$ and if it lies on $y-x+2=0$, then $\frac{9 \lambda-1}{\lambda+1}-\frac{8 \lambda+3}{\lambda+1}+2=0$ or $9 \lambda-1-(8 \lambda+3)+2(\lambda+1)=0$
$3 \lambda-2=0, \lambda=\frac{2}{3} \text { i.e. ratio is } 2: 3 \text {. }$