The ratio in which the plane $x-2y+3z=17$ divides the line joining the points $(-2,4,7)$ and $(3,-5,8)$ is:

Let the plane divide the line segment joining $P(-2,4,7)$ and $Q(3,-5,8)$ in the ratio $k:1$.The coordinates of the point of division are:
$$R=\left(\frac{3k-2}{k+1},\frac{-5k+4}{k+1},\frac{8k+7}{k+1}\right)$$
Since $R$ lies on the plane $x-2y+3z=17$:
$$\frac{3k-2}{k+1}-2\left(\frac{-5k+4}{k+1}\right)+3\left(\frac{8k+7}{k+1}\right)=17$$
$$3k-2+10k-8+24k+21=17(k+1)$$
$$37k+11=17k+17$$
$$20k=6⟹k=\frac{6}{20}=\frac{3}{10}$$
The ratio is $3:10$.