# $z_{1}$ and $z_{2}$ are two complex numbers such that $\left | z_{1}+z_{2} \right |= 4,\: \left | z_{1} \right |= 3$  and  $\left | z_{2} \right |= 2$ then $z_{1}\bar{z_{2}}+\bar{z_{1}}z_{2}$  equal Option 1) $1$ Option 2) $2$ Option 3) $3$ Option 4) $4$

D Divya Saini

$\because \left | z_{1}+z_{2} \right |^{2}=\left | z_{1} \right |^{2}+\left | z_{2} \right |^{2}+z_{1}\bar{z_{2}}+\bar{z_{1}}z_{2}$

$\Rightarrow \: z_{1}\bar{z_{2}}+\bar{z_{1}}z_{2}=16-9-4=3$

$\therefore$ Option (C)

Property of Modulus of z(Complex Number) -

$|z_{1}+z_{2}|^{2}=\left | z_{1} \right |^{2}+\left | z_{2} \right |^{2}+z_{1}.\bar{z_{2}}+z_{2}.\bar{z_{1}}$

- wherein

|.| denotes modulus of z

$\bar{z}$ denotes conjugate of z

Option 1)

$1$

This is incorrect

Option 2)

$2$

This is incorrect

Option 3)

$3$

This is correct

Option 4)

$4$

This is incorrect

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