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Question

1) For any two complex numbers z1,z2 and any real numbers a,b,|az1bz2|2+|bz1+az2|2= _________.

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Solution

Using the property,

|z1±z2|2=|z1|2+|z2|2±2Re(z1¯z2)

|az1bz2|2+|bz1+az2|2

=|az1|2+|bz2|22Re(az1× b¯z2)+

|bz1|2+|az2|2+2Re(bz1× a¯z2)

=|az1|2+|bz2|22Re(abz1¯z2)+

|bz1|2+|az2|22Re(abz1¯z2)

=(a2+b2)|z1|2+(a2+b2)|z2|2

=(a2+b2)(|z1|2+|z2|2)

Hence,
|az1bz2|2+|bz1+az2|2=(a2+b2)(|z1|2+|z2|2)


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