Prove that `|(z_1- z_2)/(1-barz_1z_2)|lt1 if |z_1|lt1,|z_2|lt1`

If z_1 and z_2 are two complex numbers such that |(barz_1-2barz_2)(2-z_1barz_2)|=1 then (A) |z_1|=1, if |z_2|!=1 (B) |z_1|=2, if |z_2|!=1 (C) |z_2|=2, if |z_1|!=1 (D) |z_2|=1, if |z_1|!=2

If z_1 and z_2 are two complex numbers for which |(z_1-z_2)(1-z_1z_2)|=1 and |z_2|!=1 then (A) |z_2|=2 (B) |z_1|=1 (C) z_1=e^(itheta) (D) z_2=e^(itheta)

if |z_1+z_2|=|z_1|+|z_2|, then prove that a r g(z_1)=a r g(z_2) if |z_1-z_2|=|z_1|+|z_2|, then prove that a r g(z_1)=a r g(z_2)=pi

(1) Re((z1z2)/(barz1))

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

Prove that |z_1+z_2|^2=|z_1|^2+|z_2|^2, ifz_1//z_2 is purely imaginary.

Prove that |z_(1) + z_(2)|^(2) + |z_(1)-z_(2)|^(2)=2|z_(1)|^(2) + 2|z_(2)|^(2)

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

For two complex numbers z_(1) and z_(2) , we have |(z_(1)-z_(2))/(1-barz_(1)z_(2))|=1 , then

For any two complex numbers z_1 and z_2 prove that: |\z_1+z_2|^2 +|\z_1-z_2|^2=2[|\z_1|^2+|\z_2|^2]