Introduction:
In this problem, we are required to find the ratio in which the line 3x + 4y - 9 = 0 divides the line segment joining the points (1,3) and (2,7).

Method:
To solve this problem, we will use the section formula.

Section Formula:
The section formula is used to find the coordinates of a point which divides a line segment into two parts in a given ratio.

Let the line segment joining the points (x1,y1) and (x2,y2) be divided in the ratio of m:n by a point (x,y). Then the coordinates of the point (x,y) are given by:

x = (nx2 + mx1) / (m + n)
y = (ny2 + my1) / (m + n)

Solution:
Given the line 3x + 4y - 9 = 0, we can write it in the form of y = mx + c as follows:

4y = -3x + 9
y = (-3/4)x + (9/4)

Let (x,y) be the point where the line 3x + 4y - 9 = 0 intersects the line segment joining (1,3) and (2,7) in the ratio m:n. Then we have:

x = (nx2 + mx1) / (m + n)
y = (ny2 + my1) / (m + n)

Substituting the values, we get:

x = (n*2 + m*1) / (m + n)
y = (n*7 + m*3) / (m + n)

Since the point (x,y) lies on the line 3x + 4y - 9 = 0, we have:

3x + 4y - 9 = 0

Substituting the value of y from the above equation, we get:

3x + 4[(n*7 + m*3) / (m + n)] - 9 = 0

Simplifying the above equation, we get:

(3n - 4m)x + 28n - 12m - 36 = 0

Since the above equation is true for all values of x, the coefficients of x and the constant term must be equal to zero. Therefore, we have:

3n - 4m = 0 (1)
28n - 12m - 36 = 0 (2)

Solving the above two equations, we get:

m:n = 7:12

Therefore, the line 3x + 4y - 9 = 0 divides the line segment joining (1,3) and (2,7) in the ratio of 7:12.

Conclusion:
Hence, we have found the ratio in which the line 3x + 4y - 9 = 0 divides the line segment joining the points (1,3) and (2,7) using the section formula.

Let the line 3x +4y-9=0 divides the lines segment joining A(1,3) and B(2,7) in ratio k:1 at point p.

therefore, coordinates of p= (2k+1/k+1,7k+3/k+1)


since, p lies on the line 3x+4y -9=0
so coordinates of p satisfies 3x +4y -9=0


=> 3(2k +1/k +1) +4(7k +3/k +1) -9 =0
=> 6k+3 +28 k +12 -9k -9=0
=> 25k +6=0

=>k= 6/24


line 3x+4y-9=0 divides the line segment joining the points A and B in 6:25 externally.


I hope it helps you :)