If z = x + iy, (x, y in R,, x neq , -1/2) , t | vedclass

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Question

The given equation is $|z|^{n}=\left(z^{2}+z\right)|z|^{n-2}+1$

$\Rightarrow \quad z^{2}+z$ is real

$\Rightarrow \quad z^{2}+z=\bar{z}^{2}+\bar{

Vedclass · Accepted Answer

$1$

Vedclass · Answer

The given equation is $|z|^{n}=\left(z^{2}+zight)|z|^{n-2}+1$

$\Rightarrow \quad z^{2}+z$ is real

$\Rightarrow \quad z^{2}+z=\bar{z}^{2}+\bar{z}$

$\Rightarrow \quad(z-\bar{z})(z+\bar{z}+1)=0$

$\Rightarrow \quad z=\bar{z}$ as $z+\bar{z}+1 
eq 0(x 
eq-1 / 2)$

Hence, the given equation reduces to

$x^{n}=x^{n}+x|x|^{n-2}+1$

$\Rightarrow \quad x|x|^{n-2}=-1$

$\Rightarrow \quad x=-1$

If $z = x + iy\, (x, y \in R,\, x \neq \, -1/2)$ , the number of values of $z$ satisfying ${\left| z \right|^n}\, = \,{z^2}{\left| z \right|^{n - 2}}\, + \,z{\left| z \right|^{n - 2}}\, + \,1\,.\,\left( {n \in N,n > 1} \right)$ is

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