Question

If the sum of the distance of point from two perpendicular lines in a plane is $1$, then prove that its locus is a square.

Answer 1

Consider that the perpendicular lines are chosen along the axes and if $\left(x,y\right)$ is the point in the first quadrant, then the equation will be,

$x+y=1$

In the same way, if the point $\left(x,y\right)$ lies in second, third and fourth quadrant, then the equations will be,

$x-y=1,-x+y=1,-x-y=1$

This shows that,

$x+y=\pm 1$ and $x-y=\pm 1$.

It can be clearly observed that the above four lines encloses a square.