Question

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

• A. a circle
• B. an ellipse
• C. a parabola
• D. a straight line

Answer 1

The correct option is D a straight line

$\left|w\right|=1$
$w=\frac{z}{z-\frac{1}{3i}}=\frac{z}{z+\frac{i}{3}}$
Taking modulus on both sides, we get
$\left|w\right|=\left|\frac{z}{z+\frac{i}{3}}\right|$
$\Rightarrow \left|z\right|=|z+\frac{i}{3}|$ ...{ $\because \left|w\right|=1$}
Substituting $z=x+iy$, we get
$6y+1=0$
Therefore, locus of z is a straight line.
Hence, option 'D' is correct.