Question

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

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

Answer 1

The correct option is A A straight line

$\mid \frac{z}{z-\frac{i}{3}}\mid =1\Rightarrow \mid z\mid =\mid z-\frac{i}{3}\mid$
Clearly locus of Z is perpendicular bisector of line joining points having complex number
$0+i0$ and $0+\frac{i}{3}.$Hence z lies on a straight line.