Prove that if [z^2 / (z-1) ] is real, then the point represented by the complex number z lies either on the real axis or on a circle passing through the origin.

Question

Prove that if  [z^2 / (z-1) ] is real, then the point represented by the complex number z lies either on the real axis or on a circle passing through the origin.

hartnn · Answer

let z = x+iy 
so, we know that 
$\large \dfrac{x^2+y^2}{(x-1)+iy}$
is REAL.
So, Equate the IMAGINARY PART of this = 0
when you multiply and divide by the conjugate of denominator,
the imaginary part has the numerator of $\large y(x^2+y^2)$
which when we equate to 0 , we get the required result :)