1.

Prove that `|z_1+z_2|^2=|z_1|^2, ifz_1//z_2`is purely imaginary.

Answer» Given
`|z_(1)+z_(2)|^(2)=|z_(1)|^(2)+|z_(2)|^(2)`
or `|z_(1)|^(2)+|z_(2)|^(2)+z_(1)bar(z)_(2)+bar(z)_(1)z_(2)=|z_(1)|^(2)+|z_(2)|^(2)`
or `z_(1)bar(z)_(2)+bar(z)_(1)z_(2)=0`
or `(z_(1))/(z_(2))+(bar(z)_(1))/(bar(z)_(2))=0`
or `(z_(1))/(z_(2))+(bar(z)_(1))/(z_(2))=0`
Hence, `(z_(1))/(z_(2))` is purely imaginary


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