For any two complex numbers `z_1` and `z_2`, prove that `|z_1+z_2|<=|z_1|+|z_2|`, `|z_1-z_2|<=|z_1|+|z_2|`, `|z+1+z_2|>=|z_1|-|z_2|` and `|z_1-z_2|>=|z_1|-|z_2|`

For any two complex numbers z_(1)" and "z_(2) prove that Im(z_(1)z_(2))=Re(z_1) Im(z_2)+ Im(z_1)Re(z_2) .

If |z_(1)+z_(2)|=|z_1|+|z_2| then

For any two complex numbers z_(1)" and "z_(2) prove that (i) Re(z_(1)z_(2))=Re(z_1) Re(z_2)-Im(z_1)Im(z_2) .

For any two complex number z_(1) and z_(2) , such that |z_(1)| = |z_(2)| = 1 and z_(1) z_(2) ne -1 , then show that (z_(1) + z_(2))/(1 + z_(1)z_(2)) is real number.

If z_1a n dz_2 are complex numbers and u=sqrt(z_1z_2) , then prove that |z_1|+|z_2|=|(z_1+z_2)/2+u|+|(z_1+z_2)/2-u|

If z_1a n dz_2 are two complex numbers and c >0 , then prove that |z_1+z_2|^2lt=(1+c)|z_1|^2+(1+c^(-1))|z_2|^2dot

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

Fill in the blanks of the following : For any two complex numbers z_1, z_2 and any real number a, b, |az_1 - bz_2|^(2) + |bz_1| + az_2|^(2) = …………

For all complex numbers z_1,z_2 satisfying |z_1|=12 and |z_2-3-4i|=5 , find the minimum value of |z_1-z_2|