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Q.
A line passing through origin and is perpendicular to two given lines $2 x+y+6=0$ and $4 x+2 y-9=0$. The ratio in which the origin divides this line, is
ManipalManipal 2016
Solution:
Equation of line perpendicular to $2 x+y+6=0$
and passes through origin is $x-2 y=0$
Now, the point of intersection of $2 x+y+6=0$
and $x-2 y=0$ is $\left(-\frac{12}{5},-\frac{6}{5}\right)$
Similarly, the point of intersection of $x-2 y=0$
and $4 x+2 y-9=0$ is $\left(\frac{9}{5}, \frac{9}{10}\right)$.
Let the origin divide the line $x-2 y=0$ in the ratio $\lambda: 1$
$\therefore x=\frac{\frac{9}{5} \lambda-\frac{12}{5}}{\lambda+1}=0$
$\Rightarrow \frac{9}{5} \lambda=\frac{12}{5}$
$\Rightarrow \lambda=\frac{12}{9}=\frac{4}{3}$