Question

If $z_1$ and $z_2$ are complex numbers with $|z_1|=|z_2|$, then which of the following is/are correct?
1. $z_1=z_2$
2. Real part of $z_1=$ Real part of $z_2$
3. Imaginary part of $z_1=$ Imaginary part of $z_2$
Select the correct answer using the statements given below :

• A. 1 only
• B. 2 only
• C. 3 only
• D. None

Answer 1

The correct option is D None

Solution:

We have, $|z_1|=|z_2|$

Let,

$z_1=x_1+i{y}_1$ and $z_2=x_2+i{y}_2$

$\because |z_1|=|z_2|$

$\therefore x_1^2+y_1^2=x_2^2+y_2^2$

or, $(x_1^2-x_2^2)+(y_1^2-y_2^2)=0$

or, $x_1^2-x_2^2=0$ and $y_1^2-y_2^2=0$

or, $x_1=\pm x_2$ and $y_1=\pm y_2$

Let, $z_1=1+i$ then $z_2=-1-i$

$|z_1|=|z_2|=\sqrt{2}$

But, $Re\left(z_1\right)\ne Re\left(z_2\right)$ and $Im\left(z_1\right)\ne Im\left(z_2\right)$ and $z_1\ne z_2$

Hence, D is the correct option.