Question 305317
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(4*0 + 2*0 - 9)/sqrt(4^2 + 2^2) = 9/sqrt(20)}}}
The distance between O(0,0) and line 2x + y + 6 = 0 is: {{{OM = absolute(2*0 + 1*0 +6)/sqrt(2^2 + 1^2) = 6/sqrt(5)}}}
Thus, OP/OQ = OM/ON = {{{(9/sqrt(20))/(6/sqrt(5))= 3/4}}}