If z is a complex number, then z.,overline z = 0 if and only if

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4-1.Complex numbers

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If $z$ is a complex number, then $z.\,\overline z = 0$ if and only if

A

$z = 0$

B

${\mathop{\rm Re}\nolimits} (z) = 0$

C

${\mathop{\rm Im}\nolimits} \,(z) = 0$

D

None of these

Solution

(a)Let $z = x + iy,\overline z = x – iy$
$\therefore $ $z\overline z = 0$

==> $(x + iy)(x – iy) = 0$

==> ${x^2} + {y^2} = 0$
It is possible only when $x$ and $y$both simultaneously zero i.e., $z = 0 + 0i = 0$

Standard 11

Mathematics

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