Question

If $z=x+iy(x,yϵR,x\ne -1/2),$ the number of values of z satisfying ${\left|z\right|}^n=z^2{\left|z\right|}^{n-2}+1.(nϵN,n>1)$is

• A. 0
• B. 1
• C. 2
• D. 3

Answer 1

The correct option is B 1

$\begin{array}{c}wewrite,\\ {\left|z\right|}^n={\left|z\right|}^{n-2}\left(z^2+z\right)+1\{weknow,allarerealno.\\ so,z^2+z={\overline{z}}^2+\overline{z}[where,z^2+zisrealno.\\ \Rightarrow z^2-{\overline{z}}^2+z-\overline{z}=0\\ \Rightarrow (z-\overline{z})(z+\overline{z}+1)=0|\begin{array}{c}z=x+iy\\ \overline{z}=x-iy\end{array}\\ \therefore 2iy(2x+1)=0\\ y=0orx=-\frac{1}{2}(Here,zisarealnumber.\\ Letsee,\\ \left|z\right|=x\\ |\begin{array}{c}{x}^n=x^{n-2}.x^2+1\\ {\left|x\right|}^{n-2}.x=-1\end{array}\\ \therefore x=-1\\ so,numberofz=1andthecorrectoptionisB.\end{array}$