Question

If $z=x+iy(x,yϵR,x\ne -1/2)$, the number of values of z satisfying $\mid z{\mid }^n=z^2\mid z{\mid }^{n-2}+z\mid z{\mid }^{n-2}+1\cdot (nϵN,n>1)$ 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
$\Rightarrow z^2+z=\stackrel{-2}{z}+\overline{z}$
$\Rightarrow (z-\overline{z})(z+\overline{z}+1)=0$
$\Rightarrow z=\overline{z}$ as $z+\overline{z}+1\ne 0(x\ne -1/2)$
Since, $z=\overline{z}$
$\Rightarrow z=\overline{z}=x$
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 number of solutions is 1.