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If $z \neq 1$ and $\frac{z^{2}}{z-1}$ is real, then the point represented by the complex number $\mathrm{z}$ lies

1) either on the real axis or on a circle not passing through the origin
2) on a circle with centre at the origin
3) either on the real axis or on a circle not passing through the origin.
4) on the imaginary axis
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(1)

Explanation

$\frac{z^{2}}{z-1}=\frac{\bar{z}^{2}}{\bar{z}-1}$

$z^{2} \bar{z}-z^{2}=z \bar{z}^{2}-\bar{z}^{2}$

$z \bar{z}(z-\bar{z})=z^{2}-\bar{z}^{2}=(z-\bar{z})(z+\bar{z})$

$z-\bar{z}=0$ or $z \bar{z}^{2}=z-\bar{z}$

$x+i y=x-i y$ or $x^{2}+y^{2}=2 x$

$y=0$, or $x^{2}+y^{2}-2 x=0$

by Diamond (75,914 points)

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