When two lines are parallel, the ratio of the distances from the origin to
their x-intercepts equals the ratio of the distances from the origin to 
their y-intercepts.  This is because, since the right triangles OBE and OCF
below are similar, 

Furthermore every line through the origin intersepted
between the two parallel lines is divided into that same ratio. 

Triangles POB and QOC are similar and thus



This is the same ratio as the ratio of the absolute values of the
x-coordinates of the two x-intesepts which is also the ratio of the
y-coordinates of the y-intercepts.

The x-intersept of the line 4x+2y=9 is (,0) and the x-intersept
of the line 2x+y=-6 is (-3,0)

The y-intersept of the line 4x+2y=9 is (0,) and the y-intersept
of the line 2x+y=-6 is (0,-6} 



And so the point O divides the segment PQ in the ratio 3:4



Edwin