Question

Find the equations of lines with slope $2$ and length of perpendicular distance from origin is equal to $\sqrt{5}$ 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 $\sqrt{5}$ units.

$\Rightarrow \frac{|2\times 0-0+c|}{\sqrt{{\left(2\right)}^2+{(-1)}^2}}=\sqrt{5}$

$\Rightarrow \frac{\left|c\right|}{\sqrt{4+1}}=\sqrt{5}$

$\Rightarrow c=\pm 5$

$\therefore$ the equations of the lines are $y=2x+5$ or $y=2x-5$