The locus of the point z satisfying ( z-1 z+1 )=k , (where k is non-zero) is

Step-by-step solution

We have, arg ( z-1 z+1 )=k,(k 0) Put z=x+i y , we get array cc & ( x-1+i y x+1+i y )=k & [ (x-1+i y)(x+1-i y) (x+1+i y)(x+1-i y) ]=k & [ (x^2-1 )+i y(2)+y^2 (x+1)^2+y^2 ]=k & [ x^2+y^2-1+2 i y (x+1)^2+y^2 ]=k & 2 y x^2+y^2-1 =k & [ (z)= ⁻¹ y x ] & (x^2+y^2 ) k-k-2 y=0 & k x^2+k y^2-2 y-k=0 & x^2+y^2- 2 y k -1=0 array Which represent a circle of centre (0,1 / k) i.e. on y -axis.