If $$\omega = \frac{z}{{z - \frac{1}{3}i}}$$ 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 $$\omega = \frac{z}{{z - \frac{1}{3}i}}$$
$$\eqalign{
& \Rightarrow \left| \omega \right| = \frac{{\left| z \right|}}{{\left| {z - \frac{1}{3}i} \right|}} = 1 \cr
& \Rightarrow \left| z \right| = \left| {z - \frac{1}{3}i} \right| \cr} $$
⇒ distance of $$z$$ from origin and point
$$\left( {0,\frac{1}{3}} \right)$$ is same hence $$z$$ lies on bisector of the line joining point $$\left( {0,0} \right)$$ 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 $${\left( {x - 1} \right)^3} + 8 = 0\,\,{\text{are}}$$