QJEEM05Ma1f73: JEE Main 2005 Mathematics - Class 11 JEEM05Ma1f73 Solution: If and , then lies on :

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.