Question
If $$n$$ is a positive integer grater than unity and $$z$$ is a complex satisfying the equation $${z^n} = {\left( {z + 1} \right)^n},$$ then
A.
$$\operatorname{Re} \left( z \right) < 2$$
B.
$$\operatorname{Re} \left( z \right) > 0$$
C.
$$\operatorname{Re} \left( z \right) = 0$$
D.
$$z$$ lies on $$x = - \frac{1}{2}$$
Answer :
$$z$$ lies on $$x = - \frac{1}{2}$$
Solution :
$$\eqalign{
& {z^n} = {\left( {z + 1} \right)^n} \cr
& \Rightarrow \,{\left| z \right|^n} = {\left| {z + 1} \right|^n}\,\,{\text{or}}\,\,\,\left| z \right| = \left| {z + 1} \right|. \cr} $$
So the distance of point $$z$$ remain same from $$\left( {0,0} \right)$$ and $$\left( {- 1,0} \right).$$
So, $$z$$ lies on perpendicular bisector of line joining $$\left( {0,0} \right)$$ and $$\left( {-1,0} \right)$$ that is on $$x = - \frac{1}{2}$$