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Question
For any two complex numbers z1 and z2 and any two real numbers a, b, find the value of \[\left| a z_1 - b z_2 \right|^2 + \left| a z_2 + b z_1 \right|^2\].
Solution
\[\left| a z_1 - b z_2 \right|^2 + \left| a z_2 + b z_1 \right|^2 = \left( a z_1 - b z_2 \right)\left( \bar{{a z_1 - b z_2}} \right) + \left( a z_2 + b z_1 \right)\left( \bar{{a z_2 + b z_1}} \right)\]
\[ = \left( a z_1 - b z_2 \right)\left( a \bar{{z_1}} - b \bar{{z_2}} \right) + \left( a z_2 + b z_1 \right)\left( a \bar{{z_2}} + b \bar{{z_1}} \right)\]
\[ = \left( a^2 z_1 \bar{{z_1}} - ab z_1 \bar{{z_2}} - ab z_2 \bar{{z_1}} + b^2 z_2 \bar{{z_2}} \right) + \left( a^2 z_2 \bar{{z_2}} + ab z_1 \bar{{z_2}} + ab z_2 \bar{{z_1}} + b^2 z_1 \bar{{z_1}} \right)\]
\[ = \left[ \left( a^2 + b^2 \right) z_1 \bar{{z_1}} + \left( a^2 + b^2 \right) z_2 \bar{{z_2}} \right]\]
\[ = \left[ \left( a^2 + b^2 \right)\left( z_1 \bar{{z_1}} + z_2 \bar{{z_2}} \right) \right]\]
\[ = \left[ \left( a^2 + b^2 \right)\left( \left| z_1 \right|^2 + \left| z_2 \right|^2 \right) \right]\]
Hence,
\[\left| a z_1 - b z_2 \right|^2 + \left| a z_2 + b z_1 \right|^2 = \left( a^2 + b^2 \right)\left( \left| z_1 \right|^2 + \left| z_2 \right|^2 \right)\]
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