Question

Find the equations of lines with slope $-1$ and length of perpendicular distance from origin is equal to $3$ units.

Answer 1

The equation of the family of lines with slope $-1$ is $y=-x+c$

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

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

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

$\Rightarrow \frac{\left|c\right|}{\sqrt{1+1}}=3$

$\Rightarrow c=\pm 3\sqrt{2}$

$\therefore$ the equations of the lines are $x-y+3\sqrt{2}=0$ or $x-y-3\sqrt{2}=0$