Question

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

• A. line
• B. parabola
• C. circle
• D. ellipse

Answer 1

The correct option is A line

Ans. $\left(a\right).$
$\left|\omega \right|=\left|\frac{z_1}{z_2}\right|=1\Rightarrow \left|z_1\right|=\left|z_2\right|$
$\left|z\right|=|z-\frac{1}{3}i|$. If $z=x+iy$, then
$\therefore x^2+y^2=x^2+{(y-\frac{1}{3})}^2$ or $-\frac{2y}{3}+\frac{1}{9}=0$
or $y=\frac{1}{6}$, i.e., a line parallel to $x$-axis.