Question

Given two complex numbers $z$ and $w$ such that $z=(1+i)w+(3-i)\overline{w}$ , where $\bar{w}$ is the conjugate of $w$ . Deduce what is $\overline{w}$ in terms of $z$ and $\overline{z}$

Answer 1

z=(1+I)w+(3-i) conjugate (w)

$z=(1+i)(3-i)\bar{w}...................(1)$

$\\\bar{z}=\overline{(1+i)w+(3-i)\bar{w}}\\$

$=\overline{(1+i)}.\overline{w}+\overline{(3-i)}.\overline{\bar{w}}\\$

$=(1-i).\overline{w}+(3+i)w........................(2)$

Multiplying (3+i) and (1+i) in equation (1) and (2), we get;

$z=(1+i)w+(3-i)\bar{w}\times(3+i)\\\bar{z}=(1-i)\bar{w}+(3+i)w\times(1+i)$

$(3+i)z=(1+i)(3+i)w+(3-i)(3+i)\bar{w}............(iii)\\(1+i)\bar{z}=(1-i)(1+i)\bar{w}+(3+i)(1+i)w............(iv)$

subtracting equation (4) from equation (3) we get

$\bar{w}[-(1+1)+(9+1)]=z(3+i)-(1+i)\bar{z}$

$\bar{w}=\frac{1}{8}[(3+i)z-(1+i)\bar{z}$