Let's denote the point of intersection of the line with the line segment AB as C.
We'll start with the equation of the line given, which is 2x + y - 4 = 0. By rearranging terms, this equation can be rewritten in the form y = -2x + 4. Therefore, the intersection point C of the line and the line segment AB satisfies the equation C_y = -2*C_x + 4.
Now, we can use the concept of vectors to express the coordinates of the point C in terms of a parameter t: C(t) = t*A + (1-t)*B. If we write this out for the x and y coordinates, we get:
1) C_x= t*A_x + (1-t)*B_x
2) C_y= t*A_y + (1-t)*B_y.
Now, we can match our second equation to C_y = -2*C_x + 4. So, we get:
-2*C_x + 4 = t*A_y + (1-t)*B_y.
Plugging in C_x from our first equation, we'll get:
-2*(t*A_x + (1-t)*B_x) + 4 = t*A_y + (1-t)*B_y.
By solving this equation, we find the value of t to be 9/11.
Knowing the value of t, we can calculate the ratio in which the line segment AB is divided by the line. The ratio is given by 1 : (1 - t) / t.
Plugging in the value of t, we find the ratio to be 1 : 9/2.
So, the line 2x + y - 4 = 0 divides the line segment AB in a 1 : 9/2 ratio or equivalently 2:9. If we take the ratio from A to B, the ratio becomes 2:9 and if from B to A, the ratio becomes 9:2.
In conclusion, the line 2x + y - 4 = 0 divides the line segment joining A(2,-2) and B(3,7) in the ratio of 2:9 (from point A to B) or 9:2 (from point B to A).