To solve the given problem, we need to find the point of intersection of the line 3x + y = 9 and the line segment joining the points (1,3) and (2,7).

Step 1: Finding the equation of the line segment
The equation of the line segment can be found using the two-point form of a straight line.

Let (x, y) be any point on the line segment. The equation of the line passing through (1,3) and (2,7) can be written as:

(y - 3)/(x - 1) = (7 - 3)/(2 - 1)
(y - 3)/(x - 1) = 4

Cross-multiplying, we get:
4(x - 1) = y - 3
4x - 4 = y - 3
4x - y = 1

So, the equation of the line segment is 4x - y = 1.

Step 2: Finding the point of intersection
To find the point of intersection, we need to solve the system of equations formed by the given line and the line segment.

Solving the system of equations:
3x + y = 9
4x - y = 1

Adding the two equations, we get:
7x = 10
x = 10/7

Substituting x = 10/7 in 3x + y = 9, we get:
3(10/7) + y = 9
30/7 + y = 9
y = 9 - 30/7
y = 63/7 - 30/7
y = 33/7

So, the point of intersection is (10/7, 33/7).

Step 3: Finding the ratio of division
To find the ratio of division, we need to calculate the distance ratio of the point of intersection from both the given points.

Using the distance formula, the distance between (1,3) and (10/7, 33/7) is given by:
d1 = √[(10/7 - 1)^2 + (33/7 - 3)^2]
= √[(3/7)^2 + (12/7)^2]
= √[(9/49) + (144/49)]
= √[(153/49)]
= √[3.122]

Similarly, the distance between (2,7) and (10/7, 33/7) is given by:
d2 = √[(10/7 - 2)^2 + (33/7 - 7)^2]
= √[(4/7)^2 + (26/7)^2]
= √[(16/49) + (676/49)]
= √[(692/49)]
= √[14.122]

Now, the ratio of division is given by:
d1/d2 = √[3.122/14.122]
= √[3/14]
= √(3/2) * √(1/7)
= √(3/2) / √(7/1)
= (√3/√2) / (√7/