For complex numbers z and w prove that `absz^2w-abs(w)^2z=z-w` if and only if z=w or `zbarw=1`

if z_1 and z_2 are two complex numbers such that absz_1lt1ltabsz_2 then prove that abs(1-z_1barz_2)/abs(z_1-z_2)lt1

If P and Q are represented by the complex numbers z_1 and z_2 such that |1/z_2+1/z_1|=|1/(z_2)-1/z_1| , then a) O P Q(w h e r eO) is the origin of equilateral O P Q is right angled. b) the circumcenter of O P Q is 1/2(z_1+z_2) c) the circumcenter of O P Q is 1/3(z_1+z_2)

Consider two complex numbers alpha and beta as alpha=[(a+b i)//(a-b i)]^2+[(a-b i)//(a+b i)]^2 , where a ,b in R and beta=(z-1)//(z+1), w here |z|=1, then find the correct statement:

If abs(z^2-1)=absz^2+1 , then z lies on

Let z_1 and z_2 be complex numbers such that z_1 nez_2 and absz_1=absz_2 if z_1 has positive real part then (z_1+z_2)/(z_1-z_2) may be

if z and w are two non-zero complex numbers such that absz=absw and argz+argw=pi , then z=

If z and w are two nonzero complex numbers such the abs(zw)=1 and arg(z)-arg(w)=pi/2 then barzw is equal to

If z_1=a+ib and z_2=c+id are complex numbers such that abs(z_1)=abs(z_2)=1 and Re(z_1barz_2)=0 , then the pair of complex numbers omega_1=a+ic and omega_2=b+id satisfies:

Find the complex number omega satisfying the equation z^3=8i and lying in the second quadrant on the complex plane.

If z_1 and z_2 be two complex numbers such that abszle1, absz_2le1, abs(z_1+iz_2)=abs(z_1-iz_2)=2 , then z_1 equals