If z is any complex number satisfying $| z - 1 | = 1$ , then which of the following is correct?

a r g (z - 1) = 2 a r g (z)

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2 a r g (z) = \frac{2}{3} a r g (z^{2} - z)

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a r g (z - 1) = a r g (z + 1)

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a r g (z) = 2 a r g (z + 1)

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Solution

The correct option is A $a r g (z - 1) = 2 a r g (z)$
Clearly, $| z - 1 | = 1$ represents a circle with centre at $(1, 0)$ and radius $1$ .
Let $P (z)$ be any point on it.
Then, $a r g (z - 1) = ∠ X C P = θ$ [say]
Therefore, $a r g (z) = ∠ X O P = \frac{θ}{2}$
Hence, $a r g (z - 1) = 2 a r g (z)$

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