Login/Sign Up
Home » If for complex numbers z1 and z2, arg (z1) – arg (z2) = 0, then show that |z1 – z2| = |z1| – |z2|.
If for complex numbers z1 and z2, arg (z1) – arg (z2) = 0, then show that |z1 – z2| = |z1| – |z2|.
CBSE | Class 11 | Complex Numbers and Quadratic Equations | Maths | NCERT Exemplar
If for complex numbers z1 and z2, arg (z1) – arg (z2) = 0, then show that |z1 – z2| = |z1| – |z2|.

Let

    \[{{z}_{1}}~=\text{ }\left| {{z}_{1}} \right|\text{ }\left( cos\text{ }{{\theta }_{1}}~+\text{ }I\text{ }sin\text{ }{{\theta }_{1}} \right)\text{ }and\text{ }{{z}_{2}}~=\text{ }\left| {{z}_{2}} \right|\text{ }\left( cos\text{ }{{\theta }_{2}}~+\text{ }I\text{ }sin\text{ }{{\theta }_{2}} \right)\]

We have,

    \[\begin{array}{*{35}{l}} arg\text{ }\left( {{z}_{1}} \right)\text{ }\text{ }-arg\text{ }\left( {{z}_{2}} \right)\text{ }=\text{ }0 \\ \Rightarrow ~{{\theta }_{1}}~\text{ }{-{\theta }_{2}}~=\text{ }0 \\ \Rightarrow ~{{\theta }_{1}}~=\text{ }{{\theta }_{2}} \\ \end{array}\]

We also have,

    \[\begin{array}{*{35}{l}} {{z}_{2}}~=\text{ }\left| {{z}_{2}} \right|\text{ }\left( cos\text{ }{{\theta }_{1}}~+\text{ }I\text{ }sin\text{ }{{\theta }_{1}} \right) \\ \Rightarrow ~{{z}_{1}}~\text{ }{-{z}_{2}}~=\text{ }\left( \left( \left| {{z}_{1}} \right|cos\text{ }{{\theta }_{1}}~-\text{ }\left| {{z}_{2}} \right|\text{ }cos\text{ }{{\theta }_{1}} \right)\text{ }+\text{ }i\text{ }\left( \left| {{z}_{1}} \right|\text{ }sin\text{ }{{\theta }_{1}}~\text{ }\left| {-{z}_{2}} \right|\text{ }sin\text{ }{{\theta }_{1}} \right) \right) \\ \end{array}\]

NCERT Exemplar Solutions For Class 11 Maths Chapter 5-21

Hence,

    \[\left| {{z}_{1}}~\text{ }{{z}_{2}} \right|\text{ }=\text{ }\left| {{z}_{1}} \right|\text{ }\text{ }\left| {{z}_{2}} \right|\]

Hence proved.

i

More Material