Given that,
$$\left| {z\omega } \right| = 1$$
$$ \Rightarrow $$ $$\left| z \right|\left| \omega \right|$$ = 1
$$ \Rightarrow $$ $$\left| z \right|$$ = $${1 \over {\left| \omega \right|}}$$
and $$Arg(z) - Arg(\omega ) = {\pi \over 2}$$
$$ \Rightarrow $$ $$Arg\left( {{z \over \omega }} \right)$$ $$= {\pi \over 2}$$
When argument of a complex number is $${\pi \over 2}$$, it means it is making an angle of $${\pi \over 2}$$ with the real axis in the counterclockwise, so it is along the imaginary axis and positive side of imaginary axis.
So, $${{z \over \omega }}$$ is a purely imaginary number that means there is no real part in this complex number.
So we can assume,
$${{z \over \omega }}$$ = $$ki$$
$$ \Rightarrow $$ $${\left| {{z \over \omega }} \right|}$$ = $$\left| {ki} \right|$$
$$ \Rightarrow $$ $${\left| {{z \over \omega }} \right|}$$ = $$\left| k \right|\left| i \right|$$
$$ \Rightarrow $$ $${\left| {{z \over \omega }} \right|}$$ = $$k$$ [ as $$\left| i \right|$$ = 1 ]
$$ \Rightarrow $$ $$\left| z \right|$$$$ \times $$$${1 \over {\left| \omega \right|}}$$ = $$k$$
$$ \Rightarrow $$ $$\left| z \right|$$$$ \times $$ $$\left| z \right|$$ = $$k$$ [ as $${1 \over {\left| \omega \right|}}$$ = $$\left| z \right|$$ ]
$$ \Rightarrow $$ $${\left| z \right|^2}$$ = $$k$$
$$ \Rightarrow $$ $$\left| z \right|$$ = $$\sqrt k $$
$$\therefore$$ $$\left| \omega \right|$$ = $${1 \over {\sqrt k }}$$
As $${{z \over \omega }}$$ is imaginary so we can write,
$${{z \over \omega }}$$ = $$ - {{\overline z } \over {\overline \omega }}$$
[ When $$z$$ is imaginary then $$z$$ = $$-\overline z $$ ]
$$ \Rightarrow $$ $$\overline z \omega $$ = $$ - z\overline \omega $$
$$ \Rightarrow $$ $$\overline z \omega $$ = $$-{{z \over \omega }}$$.$$\overline \omega $$.$$\omega $$
$$ \Rightarrow $$ $$\overline z \omega $$ = $$-{{z \over \omega }}$$.$${\left| \omega \right|^2}$$
$$ \Rightarrow $$ $$\overline z \omega $$ = $$-ki$$.$${\left( {{1 \over {\sqrt k }}} \right)^2}$$
$$ \Rightarrow $$ $$\overline z \omega $$ = $$-ki$$.$${1 \over k}$$
$$ \Rightarrow $$ $$\overline z \omega $$ = $$-i$$
Method 2 :
Given that,
$$\left| {z\omega } \right| = 1$$
$$ \Rightarrow $$ $$\left| z \right|\left| \omega \right|$$ = 1
$$ \Rightarrow $$ $$\left| z \right|$$ = $${1 \over {\left| \omega \right|}}$$
and $$Arg(z) - Arg(\omega ) = {\pi \over 2}$$
$$ \Rightarrow $$ $$Arg\left( {{z \over \omega }} \right)$$ $$= {\pi \over 2}$$
When argument of a complex number is $${\pi \over 2}$$, it means it is making an angle of $${\pi \over 2}$$ with the real axis in the counterclockwise, so it is along the imaginary axis and positive side of imaginary axis.
So, $${{z \over \omega }}$$ is a purely imaginary number that means there is no real part in this complex number.
So we can assume,
$${{z \over \omega }}$$ = $$ki$$
$$ \Rightarrow $$ $${\left| {{z \over \omega }} \right|}$$ = $$\left| {ki} \right|$$
$$ \Rightarrow $$ $${\left| {{z \over \omega }} \right|}$$ = $$\left| k \right|\left| i \right|$$
$$ \Rightarrow $$ $${\left| {{z \over \omega }} \right|}$$ = $$k$$ [ as $$\left| i \right|$$ = 1 ]
$$ \Rightarrow $$ $$\left| z \right|$$$$ \times $$$${1 \over {\left| \omega \right|}}$$ = $$k$$
$$ \Rightarrow $$ $$\left| z \right|$$$$ \times $$ $$\left| z \right|$$ = $$k$$ [ as $${1 \over {\left| \omega \right|}}$$ = $$\left| z \right|$$ ]
$$ \Rightarrow $$ $${\left| z \right|^2}$$ = $$k$$
$$ \Rightarrow $$ $$\left| z \right|$$ = $$\sqrt k $$
$$\therefore$$ $$\left| \omega \right|$$ = $${1 \over {\sqrt k }}$$
(1) Magnitude of $$\overline z \omega $$
= $$\left| {\overline z } \right|\left| \omega \right|$$
= $$\left| z \right|\left| \omega \right|$$ [ as $$\left| z \right|$$ = $$\left| {\overline z } \right|$$ ]
= $$\sqrt k $$.$${{1 \over {\sqrt k }}}$$
= 1
$$\therefore$$ The distance from the origin of $${\overline z \omega }$$ is 1.
(2) Argument of $${\overline z \omega }$$ = $$Arg\left( {\overline z \omega } \right)$$
= $$Arg\left( {\overline z } \right) + Arg\left( \omega \right)$$
= $$-Arg\left( z \right) + Arg\left( \omega \right)$$
= $$ - \left( {Arg\left( z \right) - Arg\left( \omega \right)} \right)$$
= $$ - {\pi \over 2}$$
$$\therefore$$ $${\overline z \omega }$$ is at (0, -1) on the negative side of imaginary axis and making an angle of $${\pi \over 2}$$ clockwise.
$$\therefore$$ $${\overline z \omega }$$ = 0 + (-1)$$ \times $$$$i$$ = $$-i$$