A straight line through the origin $O$ meet the parallel lines 4x + 2y = 9 and 2x + y + 6 = 0 at points P and Q respectively. The point $O$ divides the segment $PQ$ in the ratio

detailed solution

Correct option is B

It is clear that the lines lie on opposite side of the origin O. Let the equation of any line through O be  xcosθ=ysinθ. If OP=r1 If OP=r1 and OQ=r2 then the coordinates  of p are r1cosθ,r1sinθ and that of Q are -r2cosθ,-r2sinθSince P lies on 4x+2y=9,2r1(2cosθ+sinθ)=9and Q lies on 2x+y+6=0,-r2(2cosθ+sinθ)=-6so that r1r2=912=34and the required ratio is thus 3 : 4Alternately Let the equation of the line through O be  y = mx, then coordinates of P and Q arerespectively 94+2m,9m4+2m and -62+m,-6m2+m so thatOPOQ=9|4+2m|×|2+m|6=34