If arg `((z-omega)/(z-omega^2))=0` then prove that Re`(z)=-1/2(omega and omega^2` are non-real cube roots of unity).

If y_(1)=max||z-omega|-|z-omega^(2)|| , where |z|=2 and y_(2)=max||z-omega|-|z-omega^(2)|| , where |z|=(1)/(2) and omega and omega^(2) are complex cube roots of unity, then

If y_(1)=max||z-omega|-|z-omega^(2)|| , where |z|=2 and y_(2)=max||z-omega|-|z-omega^(2)|| , where |z|=(1)/(2) and omega and omega^(2) are complex cube roots of unity, then

Let S = { z: x = x+ iy, y ge 0,|z-z_(0)| le 1} , where |z_(0)|= |z_(0) - omega|= |z_(0) - omega^(2)|, omega and omega^(2) are non-real cube roots of unity. Then

Let S = { z: x = x+ iy, y ge 0,|z-z_(0)| le 1} , where |z_(0)|= |z_(0) - omega|= |z_(0) - omega^(2)|, omega and omega^(2) are non-real cube roots of unity. Then

Prove that - omega, " and " - omega^(2) are the roots of z^(2) - z + 1 = 0 , where omega " and " omega^(2) are the complex cube roots of unity .

If z_1 + z_2 + z_3 = A, z_1 + z_(2)omega+ z_(3)omega^(2)= B, z_1 + z_(2) omega^(2) + z_(3) omega=C where 1, omega, omega^2 are the cube roots of unity, then

If a=z_1+z_2+z_3, b=z_1+omega z_2+omega^2z_3,c=z_1+omega^2z_2+omegaz_3(1,omega, omega^2 are cube roots of unity), then the value of z_2 in terms of a,b, and c is (A) (aomega^2+bomega+c)/3 (B) (aomega^2+bomega^2+c)/3 (C) (a+b+c)/3 (D) (a+bomega^2+comega)/3

If a=z_1+z_2+z_3, b=z_1+omega z_2+omega^2z_3,c=z_1+omega^2z_2+omegaz_3(1,omega, omega^2 are cube roots of unity), then the value of z_2 in terms of a,b, and c is (A) (aomega^2+bomega+c)/3 (B) (aomega^2+bomega^2+c)/3 (C) (a+b+c)/3 (D) (a+bomega^2+comega)/3

If |z| = 3, the area of the triangle whose sides are z, omega z and z + omega z (where omega is a complex cube root of unity) is