The number of complex numbers z satisfying |z | vedclass

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(D) Given $|z|=1$,we can write $z = e^{i	heta}$ where $	heta \in [0, 2\pi)$.Then $\bar{z} = e^{-i	heta}$.Thus,$\frac{z}{\bar{z}} = \frac{e^{i	heta

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$8$

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(D) Given $|z|=1$,we can write $z = e^{i	heta}$ where $	heta \in [0, 2\pi)$.Then $\bar{z} = e^{-i	heta}$.Thus,$\frac{z}{\bar{z}} = \frac{e^{i	heta}}{e^{-i	heta}} = e^{i2	heta}$ and $\frac{\bar{z}}{z} = e^{-i2	heta}$.The given equation is $\left|e^{i2	heta} + e^{-i2	heta}ight| = 1$.Using Euler's formula,$e^{i2	heta} + e^{-i2	heta} = 2\cos(2	heta)$.So,$|2\cos(2	heta)| = 1$,which implies $|\cos(2	heta)| = \frac{1}{2}$.This means $\cos(2	heta) = \pm \frac{1}{2}$.For $	heta \in [0, 2\pi)$,$2	heta \in [0, 4\pi)$.If $\cos(2	heta) = \frac{1}{2}$,then $2	heta = \frac{\pi}{3}, \frac{5\pi}{3}, \frac{7\pi}{3}, \frac{11\pi}{3}$.If $\cos(2	heta) = -\frac{1}{2}$,then $2	heta = \frac{2\pi}{3}, \frac{4\pi}{3}, \frac{8\pi}{3}, \frac{10\pi}{3}$.There are $8$ distinct values for $	heta$,hence there are $8$ such complex numbers $z$.

The number of complex numbers $z$ satisfying $|z|=1$ and $\left|\frac{z}{\bar{z}}+\frac{\bar{z}}{z}\right|=1$ is:

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