Let z=x+i y and a point P represent z in the Argand plane. If the real part of z-1 z+i is 1 , then a point that lies on the locus of P is

Step-by-step solution

We have, z-1 z+i aligned & = x+i y-1 x+i y+i = (x-1)+i y x+(y+1) i & = (x-1)+i y x+(y+1) i x-(y+1) i x-(y+1) i & = x(x-1)+i x y-(x-1)(y+1) i+y(y+1) x^2+(y+1)^2 & = x(x-1)+y(y+1) x^2+(y+1)^2 + [x y-(x-1)(y+1)] i x^2+(y+1)^2 aligned Since, Re ( z-1 z+i )=1 aligned & x(x-1)+y(y+1)=x^2+(y+1)^2 & x^2-x+y^2+y=x^2+y^2+2 y+1 & -x+y=2 y+1 & x+y+1=0 & aligned (2016,-2017) lies on x+y+1=0