A variable point $z_1$ in the complex plane moves according to the condition $\left|z_1-\frac{2i}{1+i}\right|=\left|z_1\right|\left(i=\sqrt{-1}\right).$ A second variable point $z$ moves according to the condition $z_{1}z=i\omega$ where $\omega$ is imaginary cube root of unity with imaginary part positive then for the locus of $z$ which is/are correct