If ω=\;{z}{z-\;1/3 i} and |ω|=1, then z lies on [AIEEE 2005]

Correct Answer is : a straight line

Correct Answer is : a straight line

Test Index

Complex Numbers Part 1

Section: Mathematics

Question : 89 of 100

Marks: +1, -0

\omega=\;\frac{z}{z-\;\frac{1}{3} i}

|\omega|=1

z

Solution:

Given

\omega=\;\frac{z}{z-\;\frac{1}{3} i}

and

|\omega|=1

\therefore \; \frac{\left|z\right|}{\left|z-\;\frac{1}{\sqrt{3}} i\right|}=\left|\omega\right|

\Rightarrow \; \frac{\left|z\right|}{\left|z-\;\frac{1}{\sqrt{3}} i\right|}=1

\Rightarrow \left|z\right| = \left|z-\;\frac{1}{\sqrt{3}} i\right| . . . . . . . . . .

. equation (1)

\left|z\right|

represent distance of

z

from point

(0,0)

and

\left|z-\;\frac{1}{\sqrt{3}} i\right|

represent distance of

z

from point

\left(0, \;\frac{1}{\sqrt{3}}\right)

According to the equation (1) the distance of

z

from point

(0,0)

and

\left(0, \;\frac{1}{\sqrt{3}}\right)

is equal. Only if

z

is on a straight line then it will be equal distance from the both the points.

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