Question

If the roots of ${(z-1)}^{25}=2{\omega }^2{(z+1)}^{25}($ where $\omega$ is a complex cube root of unity $)$ are plotted in the argand plane, they lie in

• A. a straight line
• B. a circle
• C. an ellipse
• D. none of these

Answer 1

The correct option is B a circle

If $z$ is a root of ${(z-1)}^{25}=2{\omega }^2{(z+1)}^{25}$, then

${\left(\frac{z-1}{z+1}\right)}^{25}=2{\omega }^2\Rightarrow {\left|\frac{z-1}{z+1}\right|}^{25}=2\left|{\omega }^2\right|=2$

$\Rightarrow \left|\frac{z-1}{z+1}\right|=2^{1/25}$

As $2^{1/25}\ne 1$, we get $z$ lies on a circle.