If \[\,\omega = {z \over {z - {1 \over 3}i}}\,\] and \[\left| \omega \right| = 1\], then z lies on :
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Official solution
Given \[\,\omega = {z \over {z - {1 \over 3}i}}\,\] and \[\left| \omega \right| = 1\]
\[\therefore\] \[{{\left| z \right|} \over {\left| {z - {1 \over {\sqrt 3 }}i} \right|}} = \left| \omega \right|\]
\[ \Rightarrow \] \[{{\left| z \right|} \over {\left| {z - {1 \over {\sqrt 3 }}i} \right|}} = 1\]
\[ \Rightarrow \] \[\left| z \right| = \left| {z - {1 \over {\sqrt 3 }}i} \right|\] ..........equation (1)
\[\left| z \right|\] represent distance of z from point (0, 0) and
\[\left| {z - {1 \over {\sqrt 3 }}i} \right|\] represent distance of z from point \[\left( {0,{1 \over {\sqrt 3 }}} \right)\].
According to the equation (1) the distance of z from point (0, 0) and \[\left( {0,{1 \over {\sqrt 3 }}} \right)\] is equal. Only if z is on a straight line then it will be equal distance from the both the points.