### Complex Number Question 5

#### Question 5 - 29 January - Shift 1

For two non-zero complex number $z_1$ and $z_2$, if $Re(z_1 z_2)=0$ and $Re(z_1+z_2)=0$, then which of the following are possible ?

(A) $Im(z_1)>0$ and $Im(z_2)>0$

(B) $Im(z_1)<0$ and $Im(z_2)>0$

(C) $Im(z_1)>0$ and $Im(z_2)<0$

(D) $Im(z_1)<0$ and $Im(z_2)<0$

Choose the correct answer from the options given below :

(1) B and D

(2) B and C

(3) $A$ and $B$

(4) A and C

## Show Answer

#### Answer: (2)

#### Solution:

#### Formula: Algebra of complex numbers

$z_1=x_1+iy_1$

$z_2=x_2+iy_2$

$Re(z_1 z_2)=x_1 x_2-y_1 y_2=0$

$Re(z_1+z_2)=x_1+x_2=0$

$x_1 $ and $x_2$ are of opposite sign

$y_1 $ and $ y_2$ are of opposite sign