Question
Let $${z_1} = a + ib,{z_2} = p + iq$$ be two unimodular complex numbers such that $$\operatorname{Im} \left( {{z_1}{{\overline z }_2}} \right) = 1.$$ If $${\omega _1} = a + ip,{\omega _2} = b + iq$$ then
A.
$$\operatorname{Re} \left( {{\omega _1}{\omega _2}} \right) = 1$$
B.
$$\operatorname{Im} \left( {{\omega _1}{\omega _2}} \right) = 1$$
C.
$$\operatorname{Re} \left( {{\omega _1}{\omega _2}} \right) = 0$$
D.
$$\operatorname{Im} \left( {{\omega _1}{{\overline \omega }_2}} \right) = 1$$
Answer :
$$\operatorname{Im} \left( {{\omega _1}{{\overline \omega }_2}} \right) = 1$$
Solution :
$$\eqalign{
& \operatorname{Im} \left( {{z_1}{{\overline z }_2}} \right) = 1 \cr
& \Rightarrow \,\,bp - aq = 1 \cr
& {\omega _1}{\overline \omega _2} = \left( {a + ip} \right)\left( {b - iq} \right) = \left( {ab + pq} \right) + i\left( {bp - aq} \right) \cr
& \therefore \,\,\operatorname{Im} \left( {{\omega _1}{{\overline \omega }_2}} \right) = 1. \cr} $$