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Question
For any two complex numbers z1, z2 and any real numbers a, b, |az1 – bz2|2 + |bz1 + az2|2 = ______.
Solution
For any two complex numbers z1, z2 and any real numbers a, b, |az1 – bz2|2 + |bz1 + az2|2 = `(a^2 + b^2)(|z_1|^2 + |z2|^2)`.
Explanation:
|az1 – bz2|2 + |bz1 + az2|2
= `|az_1|^2 + |bz_2|^2 – 2 "Re"(az
_1 . b barz_2) + |bz_1|^2 + |az_2|^2 + 2 "Re" (az_1 . b barz_2)`
= `|az_1|^2 + |bz_2|^2 + |bz_1|^2 + |az_2|^2`
= `(a^2 + b^2)(|z_1|^2 + |z_2|^2)`
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