Question

If $z=x+iy(x,y\in R,x\ne -\frac{1}{2})$, the number of values of $z$ satisfying $|z{|}^n=z^2|z{|}^{n-2}+z|z{|}^{n-2}+1$, where $(n\in N,n>1)$ is

• A. $0$
• B. $1$
• C. $2$
• D. $3$

Answer 1

The correct option is B $1$

The given equation is
$|z{|}^n=(z^2+z)|z{|}^{n-2}+1$
$\Rightarrow z^2+z$ is real.
$\Rightarrow z^2+z={\overline{z}}^2+\overline{z}$
$\Rightarrow (z-\overline{z})(z+\overline{z}+1)=0$
$\Rightarrow z=\overline{z}=x$ as $z+\overline{z}+1\ne 0$
$[\because x\ne -\frac{1}{2}]$

Hence, the given equation reduces to
$x^n=x^n+x|x{|}^{n-2}+1$
$\Rightarrow x|x{|}^{n-2}=-1$
$\Rightarrow x=-1$
So, the number of solutions is $1$.