Question

Find the equations of lines with slope $3$ and length of perpendicular from the origin is $\sqrt{5}$ units.

Answer 1

The equation of the family of lines with slope $3$ is $y=3x+c$

or $3x-y+c=0$

For the required members of the family,length of perpendicular from the $(0,0)$ is $\sqrt{5}$ units.

$\Rightarrow \frac{|3\times 0-0+c|}{\sqrt{3^2+{(-1)}^2}}=\sqrt{5}$

$\Rightarrow \left|c\right|=\sqrt{5}\times \sqrt{10}=5\sqrt{2}$

$\therefore c=\pm 5\sqrt{2}$

$\therefore$ the required lines are $y=3x+5\sqrt{2}$ or $y=3x-5\sqrt{2}$