Question

lf the roots of $(z-1{)}^n=2w(z+1{)}^n$ where $n\ge 3$ and $w$ is a complex cube root of unity, are plotted in the argand plane, then these roots lie on

• A. straight line
• B. circle
• C. ellipse
• D. rectangular hyperbola

Answer 1

Answer: (B)

circle

$(z-1{)}^n=2w(z+1{)}^n$
$(\frac{z-1}{z+1}{)}^n=2w$
$|\frac{z-1}{z+1}{|}^n=|2w|=2|w|$
$|\frac{z-1}{z+1}{|}^n=2$
$|\frac{z-1}{z+1}|=2^{\frac{1}{n}}$
Now $2^{\frac{1}{n}}>1$ for $nϵN$
Therefore, $z$ lies on a circle.
Hence, option 'B' is correct.