Question 305317
A straight line thrrough the origin meets the parallel lines 4x+2y=9 and 
2x+y=-6 at points P and Q respectively. Then the point O divided the segment PQ in the ratio..
<pre><font size = 4 color = "indigo"><b> 
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, {{{(OB)/(OC)=(OE)/(EF)}}}

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

{{{(OP)/(OQ)=(OB)/(OC)=(OE)/(OF)}}}

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

{{{abs(9/4)/abs(-3)=abs(9/2)/abs(-6)=3/4}}}

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

{{{drawing(400,400,-6,6,-6,6, locate(.3,-5.5,F),
locate(1.7,2,P), locate(2.3,.6,B), locate(.3,4.7,E),
graph(400,400,-6,6,-6,6,(9-4x)/2), locate(-2,-1.5,Q),
green(line(20,15,-20,-15)), locate(-3,.6,C),
graph(400,400,-6,6,-6,6,-6-2x) )}}}

Edwin</pre>