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$z_{1}$ and $z_{2}$ are two complex numbers such that $\left |z_{1}+z_{2} \right |= 5$ , $\left |z_{1} \right |= 3$ and $\left |z_{2} \right |= 4$ , then $z_{1}\bar{z_{2}}+\bar{z_{1}}z_{2}$ equals

Option 1)
$0$

Option 2)
$1$

Option 3)
$2$

Option 4)
$3$

Answers (1)

best_answer

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

$\Rightarrow \: 25=9+16+z_{1}\bar{z_{2}}+\bar{z_{1}}z_{2}$

$\Rightarrow \: z_{1}\bar{z_{2}}+\bar{z_{1}}z_{2}=0$

$\therefore$ Option (A)

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)

$0$

This is correct

Option 2)

$1$

This is incorrect

Option 3)

$2$

This is incorrect

Option 4)

$3$

This is incorrect

Posted by

divya.saini

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