Question

If the equation $|z|(z+1{)}^8=z^8|z+1|$ where $z\in C$ and $z(z+1)\ne 0$ has distinct roots $z_1,z_2,z_3,\dots ,z_n$ (where $n\in N$ ) then which of the following is/are true?

• A. $z_1,z_2,z_3,\dots \dots ,z_n$ are concyclic points.
• B. $z_1,z_2,z_3,\dots \dots ,z_n$ are collinear points
• C. $\sum _{r=1}^n\mathrm{Re}\left(z_r\right)=\frac{-7}{2}$
• D. $\sum _{r=1}^n\mathrm{Im}\left(z_r\right)=0$

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. $\mathrm{Re}(z)=\frac{-1}{2}$. 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 $k_3$.
$\therefore \sum _{r=1}^n\mathrm{Re}\left(z_r\right)=\sum _{r=1}^7{z}_r=\frac{-1}{2}\times 7=-\frac{7}{2}$
$\text{ and }\sum _{r=1}^n\mathrm{Im}\left(z_r\right)=0+\left(k_1+k_2+k_3\right)+\left(-k_1-k_2-k_3\right)=0]$
Note: Equation (1) has two other solutions $z=0$ and $z=-1$ also, which are not acceptable as $|z(z+1)|\ne 0$ is given.]