Question

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

Answer 1

Given : Equation of lines are $4x+2y=9$ & $2x+y=6$

A line passes trough origin meets the given lines at $P$ and $Q$.

To find the ratio at which origin divides the line joining $P$ and $Q$.

Let the equation of straight line be $y=mx$

Intersection of lines :

$4x+2y=\mathrm{9..........}\left(1\right)$

$y=mx........\left(2\right)$

$2x+y=\mathrm{6...........}\left(3\right)$

$y=mx............\left(4\right)$

On putting values of $y$ from 2 to 1 and from 4 to 3.

$4x+2mx=9$

$x=\frac{9}{4+2m}y=\frac{9m}{4+2m}2x+mx=6x=\frac{6m}{2+m}y=\frac{6m}{2+m}$

Let origin divide $PQ$ in $m:1$

$P(\frac{9}{4+2m},\frac{9}{4+2m})Q(\frac{6m}{2+m},\frac{6m}{2+m})Q=\frac{m\left(\frac{9m}{4+2m}\right)+\left(\frac{6m}{2+m}\right)1}{m+1}\frac{9m}{4+2m}=-\frac{6m}{2+m}18m+9m^2=-24-12mm=-\frac{4}{3}$

Therefore origin divides $PQ$ in the ratio of $4:3$ externally.