Question

Find the equations of two straight lines which are parallel to $x+7y+2=0$ and at $\sqrt{2}$ distance away from it.

Answer 1

Equation of $1st$ line $x+4y+k=0$

Distance between them $=\sqrt{2}$

$\Rightarrow \frac{|k-2|}{\sqrt{7^2+1^2}}=\sqrt{2}$

$\Rightarrow |k-2|=\sqrt{2}\sqrt{50}=\sqrt{100}=10$

$\Rightarrow |k-2|=10$

$\Rightarrow k-2=\pm 10$

$\Rightarrow k=10+2$ $k=-10+2$

$k=12$ $k=-8$

Equation are

$x+4y+12=0$

$x+4y-8=0$