Step 1: Recall the direct ratio formula for a line cutting a segment.
There is a direct formula that gives the ratio without needing to introduce an unknown ratio variable and substitute it back into the line equation: if a line $ax + by + c = 0$ cuts the segment joining $A(x_1, y_1)$ and $B(x_2, y_2)$, then the ratio $m : n$ in which the line divides $AB$, measured from $A$ to $B$, is given by
\[ \frac{m}{n} = -\frac{a x_1 + b y_1 + c}{a x_2 + b y_2 + c} \]
This is used here as a ready made shortcut instead of building it fresh from the section formula.
Step 2: Substitute the given points into this ratio formula.
Here $a = 3$, $b = 1$, $c = -9$, and the two points are $A(1, 3)$ and $B(2, 5)$.
First find $ax_1 + by_1 + c$ using point $A$:
\[ 3(1) + 1(3) - 9 = 3 + 3 - 9 = -3 \]
Next find $ax_2 + by_2 + c$ using point $B$:
\[ 3(2) + 1(5) - 9 = 6 + 5 - 9 = 2 \]
Substitute both values into the ratio formula:
\[ \frac{m}{n} = -\frac{-3}{2} = \frac{3}{2} \]
So the line divides segment $AB$ in the ratio $3 : 2$, counted from $A$ towards $B$.
Step 3: Use the section formula with this ratio to find the point of intersection.
Let the point of intersection be $R(x, y)$, dividing $A(1, 3)$ and $B(2, 5)$ internally in the ratio $3 : 2$. By the section formula:
\[ x = \frac{3(2) + 2(1)}{3 + 2} = \frac{6 + 2}{5} = \frac{8}{5} = 1.6 \]
\[ y = \frac{3(5) + 2(3)}{3 + 2} = \frac{15 + 6}{5} = \frac{21}{5} = 4.2 \]
So $R = (1.6, 4.2)$.
Step 4: Check the point lies on the given line.
Substitute $R(1.6, 4.2)$ into $3x + y - 9$:
\[ 3(1.6) + 4.2 - 9 = 4.8 + 4.2 - 9 = 0 \]
Since this equals zero, $R$ lies on the line $3x + y - 9 = 0$, confirming the answer.
Final Answer:
\[ \boxed{\text{Ratio} = 3:2, \quad \text{Point of intersection} = (1.6, 4.2)} \]