Question

The roots of the equation $z^n=(z+1{)}^n$ on the complex plane lie on the line

• A. $2x+1=0$
• B. $2x-1=0$
• C. $x+1=0$
• D. $x-1=0$

Answer 1

The correct option is A $2x+1=0$

$|\frac{z+1}{z}{|}^n=1$
Let $n=1$,
$|z+1|=|z|$
Let $z=x+iy$
$(x+1{)}^2=x^2$
$(x+1{)}^2-x^2=0$
$(2x+1)\left(1\right)=0$
$2x+1=0$
Hence answer is A.