Step 1: Let the line divide the segment in the ratio $k : 1$ at point $P$.
Step 2: By section formula :
$$P = \left(\frac{2k + 1}{k + 1},\ \frac{5k + 3}{k + 1}\right)$$
Step 3: Since $P$ lies on $3x + y - 9 = 0$ :
$$3\left(\frac{2k+1}{k+1}\right) + \frac{5k+3}{k+1} - 9 = 0$$
Step 4: Multiply throughout by $(k + 1)$ :
$$3(2k + 1) + (5k + 3) - 9(k + 1) = 0$$
$$6k + 3 + 5k + 3 - 9k - 9 = 0 \Rightarrow 2k - 3 = 0$$
Step 5: $$k = \frac{3}{2}$$
Hence the required ratio is $3 : 2$.
Step 6: Coordinates of the point of intersection ($m : n = 3 : 2$) :
$$x = \frac{3(2) + 2(1)}{5} = \frac{8}{5},\qquad y = \frac{3(5) + 2(3)}{5} = \frac{21}{5}$$
Hence the ratio is $3 : 2$ and the point of intersection is $\left(\dfrac{8}{5},\ \dfrac{21}{5}\right)$.