Complex Numbers And Quadratic Equations question 633
Question: If $ \omega =\frac{z}{z-\frac{1}{3}i} $ and $ |\omega |=1, $ then z lies on
Options:
A) an ellipse
B) a circle
C) a straight line
D) a parabola
Show Answer
Answer:
Correct Answer: C
Solution:
As given $ \omega =\frac{z}{z-\frac{1}{3}i}\Rightarrow |\omega |=\frac{|z|}{|z-\frac{1}{3}i|}=1 $
$ \Rightarrow |z|=| z-\frac{1}{3}i | $
$ \Rightarrow $ distance of z from origin and point $ ( 0,\frac{1}{3} ) $ is same hence z lies on bisector of the line joining points $ (0,0) $ and $ (0,1/3) $ . Hence, z lies on a straight line.
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