Question

Find the equations of the lines perpendicular to the line x – y + 3 = 0 and passing at a distance numerically twice as far from the origin.

Answer 1

Given line is $x- y+3=0$

The slope of this line is $m_{1}=1$

So, the slope of the required line will be $m_{2}=-1$

Since the lines are perpendicular, the required equation is in the form of $x+y+k=0$.

The perpendicular distance from the origin is 2 units.

So,

$2=\frac{|(0)+(0)+k|}{\sqrt{1+1}} \\2=\frac{|k|}{\sqrt{2}} \\ |k|= 2 \sqrt{2} \\k=\pm 2 \sqrt{2}$

Thus, the required equation of line is $x+y\pm 2 \sqrt{2}=0.$