Prove that `Re (z_1 z_2)= Re z_1 Re z_2 - Imz_1 Imz_2` ,

Prove that Re (z_ (1) z_ (2)) = Re z_ (1) Rez_ (2) -Im z_ (1) Im z_ (2)

For any two complex numbers z_(1) and prove that quad Re(z_(1)z_(2))=Re z_(1)Re z_(2)-|mz_(1)|mz_(2)

If z_(1),z_(2) are complex number such that Re(z_(1) ) = |z_(1) - 1| , Re(z_(2)) = |z_(2) -1| and arg(z_(1) - z_(2)) = (pi)/(3) , then Im (z_(1) + z_(2)) is equal to

If z_(1)=a+ib and z_(2)=c+id are complex numbers such that Re(z_(1)bar(z)_(2))=0, then the |z_(1)|=|z_(2)|=1 and Re(z_(1)bar(z)_(2))=0, then the pair ofcomplex nunmbers omega=a+ic and omega_(2)=b+id satisfies

If z, z _1 and z_2 are complex numbers such that z = z _1 z_2 and |barz_2 - z_1| le 1 , then maximum value of |z| - Re(z) is _____.

If z_(1) , z_(2) are complex numbers such that Re(z_(1))=|z_(1)-2| , Re(z_(2))=|z_(2)-2| and arg(z_(1)-z_(2))=pi//3 , then Im(z_(1)+z_(2))=

If |z_1-1|=Re(z_1),|z_2-1|=Re(z_2) and arg (z_1-z_2)=pi/3, then Im (z_1+z_2) =

Read the following writeup carefully: If z_1 = a+ib and z_2 =c + id be two complex numbers such that |z_1| = |z_2|=1 and "Re" (z_1 bar(z_2))=0 . Now answer the following question Let k = |z_1 + z_2| + |a+ ib| , then the value of k is