Question
If $$z$$ in any complex number satisfying $$\left| {z - 1} \right| = 1,$$ then which of the following is correct ?
A.
$$arg\left( {z - 1} \right) = 2\arg z$$
B.
$$2arg\left( z \right) = \frac{2}{3}\arg \left( {{z^2} - z} \right)$$
C.
$$arg\left( {z - 1} \right) = \arg \left( {z + 1} \right)$$
D.
$$\arg z = 2\arg \left( {z + 1} \right)$$
Answer :
$$arg\left( {z - 1} \right) = 2\arg z$$
Solution :
$$\eqalign{
& {\text{Since}}\,\left| {z - 1} \right| = 1 \cr
& \therefore \,z - 1 = {e^{i\theta }},\,{\text{where}}\,\,\,\,\arg \left| {z - 1} \right| = \theta \cr
& \therefore \,z = 1 + \cos \,\theta + i\,\sin \,\theta \cr
& = \,2\cos \frac{\theta }{2}\left[ {\cos \frac{\theta }{2} + i\,\sin \,\frac{\theta }{2}} \right] \cr
& = \,2\cos \frac{\theta }{2}.{e^{\frac{{i\theta }}{2}}} = 2{\cos ^2}\frac{\theta }{2} + 2i\,\sin \frac{\theta }{2}\cos \frac{\theta }{2} \cr} $$
Thus, $$arg\left( {z - 1} \right) = 2\arg z.$$