SOLUTION: 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..

Algebra ->  Quadratic-relations-and-conic-sections -> SOLUTION: 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..      Log On


   

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..
Found 2 solutions by toidayma, Edwin McCravy:
Answer by toidayma(44) About Me  (Show Source):
You can put this solution on YOUR website!
It would be much easier if you graph it. No matter what the line through O is, according to Thales's principle, we always have: OP/OQ = OM/ON whereas OM is the the distance from O to the line 4x + 2y = 9 and ON is the distance from O to the other line. (Since the two lines are parallel, O,M and N are on a line.)
The distance between O(0,0) and line 4x + 2y -9 = 0 is: OM+=+absolute%284%2A0+%2B+2%2A0+-+9%29%2Fsqrt%284%5E2+%2B+2%5E2%29+=+9%2Fsqrt%2820%29
The distance between O(0,0) and line 2x + y + 6 = 0 is: OM+=+absolute%282%2A0+%2B+1%2A0+%2B6%29%2Fsqrt%282%5E2+%2B+1%5E2%29+=+6%2Fsqrt%285%29
Thus, OP/OQ = OM/ON = %289%2Fsqrt%2820%29%29%2F%286%2Fsqrt%285%29%29=+3%2F4

Answer by Edwin McCravy(20059) About Me  (Show Source):
You can put this solution on YOUR website!
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..
 
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, %28OB%29%2F%28OC%29=%28OE%29%2F%28EF%29

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

%28OP%29%2F%28OQ%29=%28OB%29%2F%28OC%29=%28OE%29%2F%28OF%29

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%2F4,0) 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%2F2) and the y-intersept
of the line 2x+y=-6 is (0,-6} 

abs%289%2F4%29%2Fabs%28-3%29=abs%289%2F2%29%2Fabs%28-6%29=3%2F4

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



Edwin