Question

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

• A. $1:2$
• B. $3:4$
• C. $2:1$
• D. $4:3$

Answer 1

The correct option is B

$3:4$

Explanation for the correct option:

Step 1: Finding the point P

The straight line is passing through the origin $O$, So the equation is $y=mx...\left(1\right)$

The equations are

$$\begin{gathered} 4x+2y=9...\left(2\right) \\ 2x+y=-6...\left(3\right) \end{gathered}$$

Now substitute equation $\left(1\right)$ in equation $\left(2\right)$,

$$\begin{gathered} \Rightarrow 2x+y=\frac{9}{2} \\ \Rightarrow 2x+mx=\frac{9}{2} \\ \Rightarrow \mathrm{x}(2+\mathrm{m})=\frac{9}{2} \\ \mathrm{x}=\frac{9}{2(2+\mathrm{m})} \end{gathered}$$

substitute the value of $x$ in $y$ of equation $\left(1\right)$,

$\Rightarrow y=\frac{9m}{2(2+m)}$

So the point of intersection of the line is $P\left(\frac{9}{2(2+m)},\frac{9m}{2(2+m)}\right)$

Step 2: Finding the point Q

Now substitute the equation $\left(1\right)$ in equation $\left(3\right)$,

$$\begin{gathered} \Rightarrow 2x+mx=-6 \\ \Rightarrow x(2+m)=-6 \\ \Rightarrow x=\frac{-6}{2+m} \end{gathered}$$

substitute the value of $x$ in $y$ of equation $\left(1\right)$,

$\Rightarrow y=\frac{-6m}{2+m}$

So the point of intersection of the line is $Q\left(\frac{-6}{2+m},\frac{-6m}{2+m}\right)$

Step 3: Finding the ratio

The ratio can be determined by using the section formula, $\left(\frac{m{x}_2+n{x}_1}{m+n},\frac{m{y}_2+ny1}{m+n}\right)$

where $m$and $n$ be the ratios.

Let us assume $\lambda :1$ be the ratio then, $\left(\frac{m{x}_2+n{x}_1}{m+n}\right)=0$

$$\begin{gathered} \Rightarrow \frac{{\displaystyle \frac{-6\lambda }{2+m}}+{\displaystyle \frac{9\left(1\right)}{2m+4}}}{\lambda +1}=0 \\ \Rightarrow \frac{-6\lambda }{2+m}+\frac{9}{2(m+2)}=0 \\ \Rightarrow -12\lambda +9=0 \\ \Rightarrow -12\lambda =-9 \\ \Rightarrow \lambda =\frac{3}{4} \end{gathered}$$

Thus the ratio is

$$\begin{gathered} \Rightarrow \lambda :1=\frac{3}{4}:1 \\ \lambda :1=3:4 \end{gathered}$$

Hence, option (B) $\mathbf{3}\mathbf{:}\mathbf{4}$ is the correct answer.