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

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is a part circle with centre at $\frac{ω (1 + i)}{2}$

is a part circle with radius $\frac{1}{\sqrt{2}}$

is a complete circle with radius $\sqrt{2}$

is a complete circle with centre at $\frac{ω (1 + i)}{\sqrt{2}}$

answer is C, D.

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Detailed Solution

$|\frac{i ω}{z} - \frac{2 i}{1 + i}| = |\frac{i ω}{z}| \Rightarrow |1 - - \frac{2 i}{1 + i} \times \frac{z}{i ω}| = 1$

$\Rightarrow |i ω (1 + i) - z| = \sqrt{2}$ $\Rightarrow |z - \frac{ω (1 + i)}{2}| = \frac{1}{\sqrt{2}}$

Part circle $(z \neq 0)$ , cente $= \frac{ω (1 + i)}{2}$ , rad = $\frac{1}{\sqrt{2}}$

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