Let `z` be not a real number such that `(1+z+z^2)//(1-z+z^2) in R ,` then prove tha `|z|=1.`

If z_(1) and z_(2) are two complex numbers such that |z_(1)| lt 1 lt |z_(2)| , then prove that |(1- z_(1)barz_(2))//(z_(1)-z_(2))| lt 1

If z_(1) and z_(2) are two complex number such that |z_(1)|<1<|z_(2)| then prove that |(1-z_(1)bar(z)_(2))/(z_(1)-z_(2))|<1

If z_1 and z_2 are two complex numbers such that |z_1|lt1lt|z_2| then prove that |(1-z_1barz_2)/(z_1-z_2)|lt1

If z_(1),z_(2),z_(3) are three complex numbers such that 5z_(1)-13z_(2)+8z_(3)=0, then prove that [[z_(1),(bar(z))_(1),1z_(2),(bar(z))_(2),1z_(3),(bar(z))_(3),1]]=0

Let z_(1), z_(2) be two complex numbers such that z_(1) ne 0 and z_(2)//z_(1) is purely real, then |(2i z_(1) + 5 z_(2))/(2iz_(1)-5z_(2))| is equal to

Let z_(1)=6+i&z_(2)=4-3i Let z be a a complex number such that arg((z-z_(1))/(z-z_(2)))=(pi)/(2) then z satisfies

Let z_(1),z_(2) be two complex numbers such that z_(1)+z_(2) and z_(1)z_(2) both are real, then