Question

If $n>1$, then the roots of $z^n=(z+1{)}^n$ lie on a

• A. circle
• B. straight line
• C. parabola
• D. none of thes

Answer 1

Answer: (B)

straight line

Given, $z^n={(1+z)}^n$

$\Rightarrow {\left(\frac{1+z}{z}\right)}^n=1=\cos 0+i\sin 0\Rightarrow \frac{1+z}{z}={(\cos 2\pi r+i\sin 2\pi r)}^{\frac{1}{n}}$

$\Rightarrow (\cos \frac{2\pi r}{n}+i\sin \frac{2\pi r}{n})$ where $r=0,1,2,3,..n-1$

$1+\frac{1}{z}=\cos \frac{2\pi r}{n}+i\sin \frac{2\pi r}{n}=1-2{\sin }^2\frac{r\pi }{n}+i.2\sin \frac{r\pi }{n}\cos \frac{r\pi }{n}$

$z=\frac{1}{i\left(2\sin \frac{\pi r}{n}\right)(\cos \frac{\pi r}{n}+i\sin \frac{\pi r}{n})}=\frac{1}{i\left(2\sin \frac{\pi r}{n}\right)}(\cos \frac{\pi r}{n}-i\sin \frac{\pi r}{n})$

$x+iy=-\frac{i}{2}\cot \frac{\pi r}{n}-\frac{1}{2}$

$\Rightarrow x=-\frac{1}{2}$

Hence all points lie on the line $x=-\frac{1}{2}$