Question

If $Î\pm$ is the non real $5^{\text{th }}$ root of unity and $Z_1$ and $Z_2$ are any two non zero complex numbers then find the value of $\frac{\underset{p=0}{\overset{4}{∑}}{âZ_1+{Î\pm }^P{Z}_2â}^2}{\left({Z_1âˆ\pounds }^2+{âˆ\pounds Z_2âˆ\pounds }^2\right)}$.

Answer 1

Answer: 5

${âZ_1+{Î\pm }^P{Z}_2â}^2=\left(Z_1+{Î\pm }^p{Z}_2\right)\left({\stackrel{ˉ}{Z}}_1+(\stackrel{ˉ}{Î\pm }{)}^P{\overline{Z}}_2\right)$
$={âˆ\pounds Z_1âˆ\pounds }^2+{âˆ\pounds Z_2âˆ\pounds }^2+{Î\pm }^p{\overline{Z}}_1{Z}_2+(\stackrel{ˉ}{Î\pm }{)}^P{Z}_1{\overline{Z}}_2$
Now, $\underset{p=0}{\overset{4}{∑}}{Î\pm }^p=0$ and $\underset{p=0}{\overset{4}{∑}}(\stackrel{ˉ}{Î\pm }{)}^p=0$
Hence, $\underset{p=0}{\overset{4}{∑}}{âZ_1+{Î\pm }^P{Z}_2â}^2=5\left({âˆ\pounds Z_1âˆ\pounds }^2+{âˆ\pounds Z_2âˆ\pounds }^2\right)$
$\frac{\underset{p=0}{\overset{4}{∑}}{âZ_1+{Î\pm }^P{Z}_2â}^2}{\left({âˆ\pounds Z_1âˆ\pounds }^2+{âˆ\pounds Z_2âˆ\pounds }^2\right)}=5$