If w = frac{z}{z - frac{1}{3}i} and |w| = 1,t | vedclass

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(A) Given $|w| = 1$,we have $\left| \frac{z}{z - \frac{i}{3}} \right| = 1$. This implies $|z| = |z - \frac{i}{3}|$. Let $z = x + iy$. Then $|x + iy| =

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$A$ straight line

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(A) Given $|w| = 1$,we have $\left| \frac{z}{z - \frac{i}{3}} \right| = 1$. This implies $|z| = |z - \frac{i}{3}|$. Let $z = x + iy$. Then $|x + iy| = |x + i(y - \frac{1}{3})|$. Squaring both sides,we get $x^2 + y^2 = x^2 + (y - \frac{1}{3})^2$. $x^2 + y^2 = x^2 + y^2 - \frac{2}{3}y + \frac{1}{9}$. $0 = -\frac{2}{3}y + \frac{1}{9}$ $\Rightarrow \frac{2}{3}y = \frac{1}{9}$ $\Rightarrow y = \frac{1}{6}$. This represents a horizontal straight line $y = \frac{1}{6}$ in the complex plane. Therefore,$z$ lies on a straight line.

If $w = \frac{z}{z - \frac{1}{3}i}$ and $|w| = 1$,then $z$ lies on

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