straightlines.........locus.........?

Question

the st.line x+y=1 cuts the co-ordinate axes at P&Q and a line perpendicular to this cuts axes at R&S.find the locus of intersection of the line PS & QR? [IIT]

Answer 1

P(1,0) Q(0,1)
Perpendicular x-y +k=0
R(-k,0) S(0,k)
PS x+y/k=1 so y/k= 1-x and k = y/(1-x)
QR -x/k+y= 1 so x/k= y-1 and k = x/(y-1)
y/(1-x)=x/(y-1)
y^2-y= x-x^2
x^2+y^2-x-y=0 Cf center(1/2,1/2) and radius 1/2*sqrt2