Show that the complex number `z ,`satisfying are `(z-1)/(z+1)=pi/4`lies on a circle.

Show that the complex number `z ,`satisfying are `(z-1)/(z+1)=pi/4`lie

1.	Show that the complex number `z ,`satisfying are `(z-1)/(z+1)=pi/4`lies on a circle.
Answer» Let z = x +`i`y Given that, arg` ((z - 1)/(z + 1)) = pi//4` `rArr arg(z -1) -arg (z + 1) = pi//4` `rArr arg(x +iy - 1)- arg (x + iy + 1) = pi //4` `rArr arg(x - 1 + iy)- arg ( x + 1 + iy) = pi/4` `rArr tan^(-1) (y)/(x - 1) - tan^(-1) (y)/(x + 1) = pi//4` `rArr tan^(-1)[((y)/(x -1)-(y)/(x +1))/ (1 +((y)/(x -1))((y)/(x+1)))]=pi//4` `rArr [y[(x + 1- x +1)/(x^(2) - 1)]]/((x ^(2)-1 + Y^(2))/(x^(2)-1))= tan pi //4` `rArr (2y)/(x^(2) + Y^(2) - 1) = 1` `rArr x^(2) + y^(2) - 1 = 2 y` `rArr x^(2) + Y^(2) - 1 = 0` which represents a circle .