Numbers of complex numbers z, such that `abs(z)=1`
and `abs((z)/bar(z)+bar(z)/(z))=1` is

The number of complex numbers z such that |z-1|=|z+1|=|z-i| is

Let z,z_(0) be two complex numbers. It is given that abs(z)=1 and the numbers z,z_(0),zbar_(0),1 and 0 are represented in an Argand diagram by the points P, P_(0) ,Q,A and the origin, respectively. Show that /_\POP_(0) and /_\AOQ are congruent. Hence, or otherwise, prove that abs(z-z_(0))=abs(zbar(z_(0))-1)=abs(zbar(z_(0))-1) .

If z and we are two complex numbers such that |zw|=1 and arg(z)-arg(w)=(pi)/(2) then show that barzw=-i

z_(1) and z_(2) are two complex number such that |z_(1)|=|z_(2)| and arg (z_(1))+arg(z_(2))=pi , then show that z_(1)=-barz_(2)

State true of false for the following: If z is a complex number such that z ne0 and Re(z)=0 then 1_(m)(z^(2))=0

For complex number z =3 -2i,z + bar z = 2i Im(z) .

For complex number z = 5 + 3i value z -barz=10 .

State true of false for the following: Let z_(1) and z_(2) be two complex number's such that |z_(1)+z_(2)|=|z_(1)|+|z_(2)| then arg (z_(1)-z_(2))=0

If z_(1), z_(2) and z_(3), z_(4) are two pairs of conjugate complex numbers, then find arg ((z_(1))/(z_(4)))+arg((z_(2))/(z_(3))) .