If $$\omega = \frac{z}{{z - \frac{1}{3}i}}\,{\text{and }}\left| \omega \right| = 1,$$ then $$z$$ lies on
A.
an ellipse
B.
a circle
C.
a straight line
D.
a parabola
Answer :
a straight line
Solution :
As given $$w = \frac{z}{{z - \frac{1}{3}i}}$$
$$ \Rightarrow \,\,\left| w \right| = \frac{{\left| z \right|}}{{\left| {z - \frac{1}{3}i} \right|}} = 1$$
⇒ distance of $$z$$ from origin and point $$\left( {0,\frac{1}{3}} \right)$$ is same hence $$z$$ lies on bisector of the line joining points (0, 0) and $$\left( {0,\frac{1}{3}} \right).$$
Hence $$z$$ lies on a straight line.
Releted MCQ Question on Algebra >> Complex Number
Releted Question 1
If the cube roots of unity are $$1,\omega ,{\omega ^2},$$ then the roots of the equation $${\left( {x - 1} \right)^3} + 8 = 0\,\,{\text{are}}$$