Question

Find the equations of lines with slope $-2$ and length of perpendicular distance from origin is equal to $\sqrt{3}$ units.

Answer 1

The equation of the family of lines with slope $-2$ is $y=-2x+c$

or $2x-y-c=0$ .......$\left(1\right)$

For the required members of the family, the perpendicular distance from the origin $(0,0)$ is $\sqrt{3}$ units.

$\Rightarrow \frac{|2\times 0-0-c|}{\sqrt{{\left(2\right)}^2+{(-1)}^2}}=\sqrt{3}$

$\Rightarrow \frac{\left|c\right|}{\sqrt{4+1}}=\sqrt{6}$

$\Rightarrow c=\pm \sqrt{30}=\pm \sqrt{30}$

$\therefore$ the equations of the lines are $2x-y+\sqrt{30}=0$ or $2x-y-\sqrt{30}=0$