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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
$0$
$1$
$2$
$3$
Solution
The given equation is $|z|^{n}=\left(z^{2}+z\right)|z|^{n-2}+1$
$\Rightarrow \quad z^{2}+z$ is real
$\Rightarrow \quad z^{2}+z=\bar{z}^{2}+\bar{z}$
$\Rightarrow \quad(z-\bar{z})(z+\bar{z}+1)=0$
$\Rightarrow \quad z=\bar{z}$ as $z+\bar{z}+1 \neq 0(x \neq-1 / 2)$
Hence, the given equation reduces to
$x^{n}=x^{n}+x|x|^{n-2}+1$
$\Rightarrow \quad x|x|^{n-2}=-1$
$\Rightarrow \quad x=-1$
Similar Questions
Let $z$ be complex number satisfying $|z|^3+2 z^2+4 z-8=0$, where $\bar{z}$ denotes the complex conjugate of $z$. Let the imaginary part of $z$ be nonzero.
Match each entry in List-$I$ to the correct entries in List-$II$.
| List-$I$ | List-$II$ |
| ($P$) $|z|^2$ is equal to | ($1$) $12$ |
| ($Q$) $|z-\bar{z}|^2$ is equal to | ($2$) $4$ |
| ($R$) $|z|^2+|z+\bar{z}|^2$ is equal to | ($3$) $8$ |
| ($S$) $|z+1|^2$ is equal to | ($4$) $10$ |
| ($5$) $7$ |
The correct option is:
