For two non–zero complex number z1 and z2, if Re(z1z2)=0 and Im(z1+z2)=0 then which of the following are possible?A) Re(z1)>0 and Re(z2)>0 B) Re(z1) 0C)Re(z1)>0 and Re(z2)<0 D) Re(z1)<0 and Re(z2)<0 Choose the correct answer from the options given below

Re(z1z2)=0,Im(z1+z2)=0 z1=x1+iy1,z2=x2+iy2 y1+y2=0 and x1x2−y1y2=0 y1=−y2 and x1x2=y1y2 x1x2<0 ⇒x1,x2 are of opposite signy1,y2 are of opposite sign

For two non–zero complex number z1 and z2, if Re(z1z2)=0 and Im(z1+z2)=0 then which of the following are possible?A) Re(z1)>0 and Re(z2)>0 B) Re(z1) 0C)Re(z1)>0 and Re(z2)<0 D) Re(z1)<0 and Re(z2)<0 Choose the correct answer from the options given below

Re(z1z2)=0,Im(z1+z2)=0 z1=x1+iy1,z2=x2+iy2 y1+y2=0 and x1x2−y1y2=0 y1=−y2 and x1x2=y1y2 x1x2<0 ⇒x1,x2 are of opposite signy1,y2 are of opposite sign

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For two non–zero complex number $z_{1}$ and $z_{2},$ if $Re (z_{1} z_{2}) = 0$ and $Im (z_{1} + z_{2}) = 0$ then which of the following are possible?
A) $Re (z_{1}) > 0$ and $Re (z_{2}) > 0$ B) $Re (z_{1}) < 0$ and $Re (z_{2}) > 0$
C) $Re (z_{1}) > 0$ and $Re (z_{2}) < 0$ D) $Re (z_{1}) < 0$ and $Re (z_{2}) < 0$
Choose the correct answer from the options given below

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B and D

B and C

A and B

A and C

answer is B.

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Detailed Solution

$Re (z_{1} z_{2}) = 0, Im (z_{1} + z_{2}) = 0$
$z_{1} = x_{1} + i y_{1}, z_{2} = x_{2} + i y_{2}$
$y_{1} + y_{2} = 0$ and $x_{1} x_{2} - y_{1} y_{2} = 0$
$y_{1} = - y_{2}$ and $x_{1} x_{2} = y_{1} y_{2}$
$x_{1} x_{2} < 0$
$\Rightarrow x_{1}, x_{2}$ are of opposite sign
$y_{1}, y_{2}$ are of opposite sign