A straight line through origin meets parallel lines $4x+2y=9$ and $2x+y+6=0$ at P and Q. Find PQ segment ratio by origin Let the equation of line be $y=mx$

Then 
$$2x+y=\frac 92$$
$$2x+mx=\frac 92$$
$$x=\frac{9}{2(2+m)}$$

$$y=\frac{9m}{2(2+m)}$$for the second line 
$$x=\frac{-6}{m+2}$$
$$y=\frac{-6m}{m+2}$$
Let (0,0) divide it in the ratio k:1
$$0=k\frac{9}{2(m+2)}+\frac{-6}{m+2}$$
$$k=\frac 43$$
But the answer given is $\frac 34$. 

Now this isn’t simply a perspective problem, because both options 4/3 and 3/4 were present. How do we tell which one is right?
Note that the question refers to P and $4x+2y=9$ first, and Q and $2x+y +6=0$ second. So, it is logical to assume that the ratio refers to $\frac{OP}{OQ}$. 

Also, the ratio can be conveniently found as the ratio of the distances $d_P$ and $d_Q$ between the origin and the two parallel lines, because the distance line, PQ and the two parallel lines form a pair of similar right triangles. Thus,

$$\frac{OP}{OQ}=\frac{d_P}{d_Q}=\frac{\frac{|0+0-9|}{\sqrt{4^2+2^2}}}{\frac{|0+0+6|}{\sqrt{2^2+1^2}}}=\frac{\frac92}6=\frac34$$
First, do you see that the origin lies between P and Q?  It can't divide PQ in ratio "4/3" because that is larger than 1 and implies that that the origin is NOT in the interval.