If `z_1andz_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`

Let z_(1) and z_(2) be two complex numbers such that |z_(1)+z_(2)|^(2)=|z_(1)|^(2)+|z_(2)|^(2) . Then,

If z_(1), z_(2), z_(3) are three complex numbers in A.P., then they lie on

If z_(1), z_(2) are two complex numbers such that Im(z_(1)+z_(2))=0 and Im(z_(1) z_(2))=0 , then

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

If z_(1),z_(2) are two complex numbers satisfying the equation : |(z_(1)-z_(2))/(z_(1)+z_(2))|=1 , then (z_(1))/(z_(2)) is a number which is

If z_(1) and z_(2) are two non-zero complex numbers such that |z_(1)+z_(2)|=|z_(1)|+|z_(2)| , then arg. z_(1)- arg. z_(2) equals :

Let z_1, z_2 be two complex numbers represented by points on the circle |z_1|= 1 and and |z_2|=2 are then

If z is a complex number such that |z|=1, z ne 1 , then the real part of |(z-1)/(z+1)| is

The number of complex numbers z such that |z-1|=|z+1|=|z-i| equals

If z_(1) and z_(2) are complex number and (z_(1))/(z_(2)) is purely imaginary then |((z1+z2)/(z1-z2))|=