Question

If $z=2+i$, then $\left(z-1\right)\left(\bar{z}-5\right)+\left(\bar{z}-1\right)\left(z-5\right)$ is equal to

• A. $2$
• B. $7$
• C. $-1$
• D. $-4$

Answer 1

Answer: (D)

$\left(z-1\right)\left(\bar{z}-5\right)+\left(\bar{z}-1\right)\left(z-5\right)$
$=2Re\left[\left(z-1\right)\left(\bar{z}-5\right)\right]$
$\left(\because\,z_{1}\bar{z}_{2}+z_{2}\bar{z}_{1}=2Re\left(z_{1}\bar{z}_{2}\right)\right)$
$=2 Re \left[\left(1+i\right)\left(-3-i\right)\right]=2\left(-2\right)=-4\,\,\,\,$ (Given $z = 2 +i$)