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Standard XII

Mathematics

Complex Numbers

If z1 and ...

Question

If $z_{1}$ and $z_{2}$ are complex numbers, prove that ${| z_{1} + z_{2} |}^{2} = {| z_{1} |}^{2} + {| z_{2} |}^{2}$ if and only is $z_{1} {¯ z}_{2}$ is pure imaginary.

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Solution

${| z_{1} + z_{2} |}_{1}^{2} = {| z_{1} |}_{1}^{2} + {| z_{2} |}_{2}^{2} + 2 R e (z_{1} {¯ ¯ ¯ z}_{2})$
If $z_{1} {¯ ¯ ¯ z}_{2}$ purely imaginary, then $R e$ $(z_{1} {¯ ¯ ¯ z}_{2}) = 0$
$∴ {| z_{1} + z_{2} |}_{1}^{2} = {| z_{1} |}_{1}^{2} + {| z_{2} |}_{2}^{2}$ ...... $(2)$
Conversely if $(1)$ holds, then $R e$ $(z_{1} {¯ ¯ ¯ z}_{2}) = 0$
It means that $z_{1} {¯ ¯ ¯ z}_{2}$ is purely imaginary.

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If z1 and z2 are complex numbers, prove that |z1+z2|2=|z1|2+|z2|2 if and only is z1 ¯z2 is pure imaginary.

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If $z_{1}$ and $z_{2}$ are complex numbers, prove that ${| z_{1} + z_{2} |}^{2} = {| z_{1} |}^{2} + {| z_{2} |}^{2}$ if and only is $z_{1} {¯ z}_{2}$ is pure imaginary.