If a complex number z satisfies |z^2-1 |=|z|^2+1 , then z lies on

Step-by-step solution

Given, |z^2-1 |=|z|^2+1 Let z=x+i y aligned & |(x+i y)^2-1 |=|x+i y|^2+1 & |x^2-y^2+2 i x y-1 |= (x^2+y^2 )+1 & | (x^2-y^2-1 )+2 i x y |= (x^2+y^2+1 ) & (x^2-y^2-1 )^2+4 x^2 y^2 =x^2+y^2+1 & (x^2-y^2 )^2+1-2 (x^2-y^2 )+4 x^2 y^2 & = (x^2+y^2+1 )^2 & =x^4+y^4+2 x^2 y^2+1+2 x^2+2 y^2 & -2 x^2 y^2-2 x^2+4 x^2 y^2=2 x^2 y^2+2 x^2 & -2 x^2=2 x^2 & 4 x^2=0 x=0 & z=x+i y=0+i y & z=i y (x, y)=(0, y) & aligned Hence, z lies on the imaginary axis.