If `z,z_1,z_2inCC` then (vii)`|z_1+z_2|^2=|z_1|^2+|z_2|^2+2Re(z_1barz_2)` (viii)`|z_1-z_2|^2=|z_1|^2+|z_2|^2-2Re(z_1barz_2)` (ix)`|z_1+z_2|^2+|z_1-z_2|^2=2(|z_1|^2+|z_2|^2)` (x) `|az_1-bz_2|^2+|bz_1+az_2|^2=(a^2+b^2)(|z_1|^2+|z_2|^2)` where `a,b in RR`

Prove that |z_1+z_2|^2+|z_1-z_2|^2 =2|z_1|^2+2|z_2|^2 .

Prove that |1-barz_1z_2|^2-|z_1-z_2|^2=(1-|z_1|^2)(1-|z_2|^2) .

If |z_1|=1,|z_2|=1 then prove that |z_1+z_2|^2+|z_1-z_2|^2 =4.

If z_1ne-z_2 and |z_1+z_2|=|1/z_1 + 1/z_2| then :

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

If |z_1+z_2|=|z_1-z_2| and |z_1|=|z_2|, then (A) z_1=+-iz_2 (B) z_1=z_2 (C) z_=-z_2 (D) z_2=+-iz_1

Let z_1=r_1(costheta_1+isintheta_1)a n dz_2=r_2(costheta_2+isintheta_2) be two complex numbers. Then prove that |z_1+z_2|^2=r1 2+r2 2+2r_1r_2cos(theta_1-theta_2) or |z_1+z_2|^2=|z_1|^2+|z_2|^2+2|z_1||z_2|^()_cos(theta_1-theta_2) |z_1-z_2|^2=r1 2+r2 2-2r_1r_2cos(theta_1-theta_2) or |z_1-z_2|^2=|z_1|^2+|z_2|^2-2|z_1||z_2|^()_cos(theta_1-theta_2)

If z_1 , z_2 are nonreal complex and |(z_1+z_2)/(z_1-z_2)| =1 then (z_1)/(z_2) is

If kgt1,|z_1|,k and |(k-z_1barz_2)/(z_1-kz_2)|=1 , then (A) z_2=0 (B) |z_2|=1 (C) |z_2|=4 (D) |z_2|ltk

If z,z=z_(2) and |z_(1)+z_(2)|=|(1)/(z_(1))+(1)/(z_(2))| then