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

Question

If z is any complex number satisfying |z-1| =1 , then which of the following is correct? (A)  arg (z-1)=2 arg (z)  (B)  2 arg (z)=(2/3) arg (z2-z)  (C)  arg (z-1)= arg (z+1)  (D)  arg (z)=2 arg (z+1)

Tardigrade Admin · Accepted Answer

Clearly, |z-1| =1 represents a circle with centre at (1, 0) and radius 1. Let P(z) be any point on it. Then, arg (z-1)= angle XCP=θ [say] ∴ arg (z)= angle XOP=(θ /2) Hence, arg (z-1)=2 arg (z)

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

$ar g (z - 1) = 2 ar g (z)$

$2 ar g (z) = \frac{2}{3} ar g (z^{2} - z)$

$ar g (z - 1) = ar g (z + 1)$

$ar g (z) = 2 ar g (z + 1)$

Clearly, $∣ z - 1∣ = 1$ represents a circle with centre at $(1, 0)$ and radius 1.
Let $P (z)$ be any point on it.
Then, $ar g (z - 1) = ∠ XCP = θ$ [say]
$∴$ $ar g (z) = ∠ XOP = \frac{θ}{2}$
Hence, $ar g (z - 1) = 2 ar g (z)$

Q. If z is any complex number satisfying ∣z−1∣=1 , then which of the following is correct?

arg(z−1)=2arg(z)

2arg(z)=32​arg(z2−z)

arg(z−1)=arg(z+1)

arg(z)=2arg(z+1)