(A) Let $z = x + iy$,where $x, y \in \mathbb{R}$. Then $\overline{z} = x - iy$.Given $z \cdot \overline{z} = 0$.Substituting the values,we get $(x + i

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(A) Let $z = x + iy$,where $x, y \in \mathbb{R}$. Then $\overline{z} = x - iy$.Given $z \cdot \overline{z} = 0$.Substituting the values,we get $(x + iy)(x - iy) = 0$.This simplifies to $x^2 - (iy)^2 = 0$,which is $x^2 + y^2 = 0$.Since $x^2 \ge 0$ and $y^2 \ge 0$ for all real $x$ and $y$,the sum $x^2 + y^2 = 0$ is possible if and only if $x = 0$ and $y = 0$.Therefore,$z = 0 + 0i = 0$.

Class 11 Mathematics 4-1.Complex numbers Conjugate and Modulus of complex numbers

Medium

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

A
$z = 0$
B
$\text{Re}(z) = 0$
C
$\text{Im}(z) = 0$
D
$\text{None of these}$

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4-1.Complex numbers

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Conjugate and Modulus of complex numbers

Let $z_1$ and $z_2$ be two complex numbers such that $z_1 + z_2$ and $z_1 z_2$ are both real. Then:

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If $z = r{e^{i\theta }},$ then $|{e^{iz}}| = $

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If $z \neq 0$ is a complex number such that $|z - \frac{1}{z}| = 2$,then the maximum value of $|z|$ is:

DifficultJEE MAIN 2022

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If $\alpha, \beta$ are non-zero integers and $z=(\alpha+i \beta)(2+7 i)$ is a purely imaginary number,then the minimum value of $|z|^2$ is

MediumTS EAMCET 2023

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

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