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

Question

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?

Answer 1

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,

$$\frac{OP}{OQ}=\frac{d_P}{d_Q}=\frac{\frac{|0+0-9|}{\sqrt{4^2+2^2}}}{\frac{|0+0+6|}{\sqrt{2^2+1^2}}}=\frac{\frac92}6=\frac34$$

Answer 2

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.

Answer 3

Technically, both are correct. The given answer is ${3\over 4}$ because most textbooks assume the ration to be $1:k$ not $k:1$