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 omega is the nth root of unity and Z_1 , Z_2 are any two complex numbers , then prove that . Sigma_(k=0)^(n-1)| z_1+ omega^k z_2|^2=n{|z_1|^2+|z_2|^2} where n in N

If z_1 and z_2 , are two non-zero complex numbers such tha |z_1+z_2|=|z_1|+|z_2| then arg(z_1)-arg(z_2) is equal to

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)

Consider z_(1)andz_(2) are two complex numbers such that |z_(1)+z_(2)|=|z_(1)|+|z_(2)| Statement -1 arg (z_(1))-arg(z_(2))=0 Statement -2 The complex numbers z_(1) and z_(2) are collinear.

State true of false for the following: Let z_(1) and z_(2) be two complex number's such that |z_(1)+z_(2)|=|z_(1)|+|z_(2)| then arg (z_(1)-z_(2))=0

If the complex numbers z_(1) and z_(2) arg (z_(1))- arg(z_(2))=0 then showt aht |z_(1)-z_(2)|=|z_(1)|-|z_(2)| .

If z is a complex number such that |z|>=2 then the minimum value of |z+1/2| is

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

If z_1,z_2 and z_3,z_4 are two pairs of conjugate complex numbers then arg(z_1/z_4)+arg(z_2/z_3)=

z_(1) and z_(2) are two complex number such that |z_(1)|=|z_(2)| and arg (z_(1))+arg(z_(2))=pi , then show that z_(1)=-barz_(2)