Question

If the equation $| z |( z +1)^8= z ^8| z +1|$ where $z \in C$ and $z ( z +1) \neq 0$ has distinct roots $z _1, z _2, z _3, \ldots, z _{ n }$ (where $n \in N$ ) then which of the following is/are true?

• A. $z _1, z _2, z _3, \ldots \ldots, z _{ n }$ are concyclic points.
• B. $z _1, z _2, z _3, \ldots \ldots, z _{ n }$ are collinear points
• C. $ \displaystyle\sum_{ r =1}^{ n } \operatorname{Re}\left( z _{ r }\right)=\frac{-7}{2}$
• D. $\displaystyle\sum_{r=1}^n \operatorname{Im}\left(z_r\right)=0$

Answer 1

Answer: (B), (C), (D)

We have $| z |( z +1)^8= z ^8| z +1|$....(1)
Taking modulus on both sides, we get
$ | z || z +1|^8=| z |^8| z +1| $
$\therefore | z +1|^7=| z |^7 $
$\Rightarrow | z +1|=| z |$
which represents the locus of $z$ will be a straight line which is perpendicular bisector of the line segment joining $(-1,0)$ and $(0,0)$ i.e. $\operatorname{Re}(z)=\frac{-1}{2}$. Also there will be exactly 7 distinct ' $z$ ' satisfying given equation, i.e. $ z =\frac{-1}{2}, \frac{-1}{2} \pm ki$ where $k$ has 3 distinct positive values $k _1, k _2$ and $k _3$.
$\therefore \displaystyle\sum_{ r =1}^{ n } \operatorname{Re}\left( z _{ r }\right)=\displaystyle\sum_{ r =1}^7 z _{ r }=\frac{-1}{2} \times 7=-\frac{7}{2} $
$\text { and } \left.\displaystyle\sum_{ r =1}^{ n } \operatorname{Im}\left( z _{ r }\right)=0+\left( k _1+ k _2+ k _3\right)+\left(- k _1- k _2- k _3\right)=0\right]$
Note: Equation (1) has two other solutions $z=0$ and $z=-1$ also, which are not acceptable as $|z(z+1)| \neq 0$ is given.]