The line x+y = 1 cuts the coordinate axes at P and Q and a line perpendicular to it meet the axes in R and S. The equation to the locus of the point of intersection of the lines PS and QR is

The line x+y=1 meets the coordinate axes at P(1, 0), Q(0, 1) 

P and S are on y-axis ,Q and R are on x-axis

Then the point of intersection is origin ( given they are perpendicular)

hence P and Q become the ends of diameter (angle in a semi circle is right angle)

$\therefore \left(x-1\right)\left(x-0\right)+\left(y-0\right)\left(y-1\right)=0$

$\Rightarrow x^2+y^2-x-y=0$