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) = 2arg (z + 1)

Answer 1

Answer: (A)

arg (z – 1) = 2 arg z

Answer 2

Sol. Let z – 1 = e^iq (given |z – 1| = 1)

z = 1 + cos q + i sin q

= 2 cos² $\frac{\theta}{2}$ + 2i sin $\frac{\theta}{2}$ cos $\frac{\theta}{2}$

= 2 cos $\left( \cos\frac{\theta}{2} + i\sin\frac{\theta}{2} \right)$.

\ arg z = $\frac{\theta}{2}$ = $\frac{1}{2}$ arg (z – 1) or arg (z – 1) = 2 arg z.