If `z_1,z_2inC`, prove that `(z_1+z_2)^2=Z_1^2 + 2z_1z_2 + zunderset(2)overset(2)`

If z_1,z_2 in C , show that (z_1+ z_2)^2= z_1^2+2 z_1 z_2+ z_2^2 .

For any two complex numbers z_1 and z_2 , prove that Re (z_1 z_2) = Re z_1 Re z_2 - Imz_1 Imz_2

If z_1= 2 -i, z_2= 1+ i , find |(z_1+z_2+1)/(z_1-z_2+1)| .

If a x_1^2+by_1^2+c z_1^2=a x_2^2+b y_2^2+c z_2^2=a x_3 ^2+b y_3 ^2+c z_3^ 2=d a x_2x_3+b y_2y_3+c z_2z_3=a x_3x_1+b y_3y_1+c z_3z_1= a x_1x_2+b y_1y_2+c z_1z_2=f, then prove that | x_1 y_1 z_1 x_2 y_2 z_2 x_3 y_3 z_3 |= (d-f){(d+2f)/(a b c)}^(1//2)

If z_1 and z_2 are two complex number such that |(z_1-z_2)/(z_1+z_2)|=1 , Prove that iz_1/z_2=k where k is a real number Find the angle between the lines from the origin to the points z_1 + z_2 and z_1-z_2 in terms of k

If z_1 =-3i, z_2= 3 +4i and z_3=2- 3i , verify that z_1(z_2+z_3)=z_1z_2+z_1z_3 .

Define addition and multiplication of two complex numbers z_1 and z_2 . Hence show that : I_m (z_1 z_2)=R_e (z_1)I_m (z_2)- I_m (z_1)R_e(z_2) .

If z_1 and z_2 (ne 0) are two complex numbers, prove that : |z_1 z_2|= |z_1||z_2| .