Question
If $${z_1} \ne - {z_2}$$ and $$\left| {{z_1} + {z_2}} \right| = \left| {\frac{1}{{{z_1}}} + \frac{1}{{{z_2}}}} \right|$$ then
A.
at least one of $${z_1},{z_2}$$ is unimodular
B.
both $${z_1},{z_2}$$ are unimodular
C.
$${z_1} \cdot {z_2}$$ is unimodular
D.
None of these
Answer :
$${z_1} \cdot {z_2}$$ is unimodular
Solution :
$$\eqalign{
& \left| {{z_1} + {z_2}} \right| = \left| {\frac{{{z_1} + {z_2}}}{{{z_1}{z_2}}}} \right| \cr
& {\text{or, }}\left| {{z_1} + {z_2}} \right|\left( {1 - \frac{1}{{\left| {{z_1}{z_2}} \right|}}} \right) = 0 \cr
& \therefore \,\,\left| {{z_1}{z_2}} \right| = 1. \cr} $$