Question

If $w=\frac{z}{z-\frac{1}{3}i}$ 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

$\begin{array}{c}Given\\ \omega =\frac{3z}{3z-i}\therefore \left|\omega \right|=\frac{3\left|z\right|}{\left|3z-i\right|}\\ \Rightarrow \left|3z-i\right|=3\left|z\right|\\ \Rightarrow \left|3\left(x\right)+i\left(3y-1\right)\right|=\left|3\left(x+iy\right)\right|\left(z=x+iy\right)\\ \Rightarrow {\left(3x\right)}^2+{\left(3y-1\right)}^2=9\left(x^2+y^2\right)\\ \Rightarrow 6y-1=0\\ Whichisstraightline\\ Hence,optionDisthecorrectanswer.\end{array}$