Question

If $\alpha$ 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 $\frac{\displaystyle\sum_{ p =0}^4\left| Z _1+\alpha^{ P } Z _2\right|^2}{\left(\left. Z _1\right|^2+\left| Z _2\right|^2\right)}$.

Answer 1

$\left|Z_1+\alpha^{ P } Z_2\right|^2=\left(Z_1+\alpha^{ p } Z _2\right)\left(\bar{Z}_1+(\bar{\alpha})^{ P } \overline{ Z }_2\right) $
$=\left| Z _1\right|^2+\left| Z _2\right|^2+\alpha^{ p } \overline{ Z }_1 Z _2+(\bar{\alpha})^{ P } Z _1 \overline{ Z }_2$
Now, $\displaystyle\sum_{p=0}^4 \alpha^p=0$ and $\displaystyle\sum_{p=0}^4(\bar{\alpha})^p=0$
Hence, $\displaystyle\sum_{ p =0}^4\left| Z _1+\alpha^{ P } Z _2\right|^2=5\left(\left| Z _1\right|^2+\left| Z _2\right|^2\right)$
$\frac{\displaystyle\sum_{ p =0}^4\left| Z _1+\alpha^{ P } Z _2\right|^2}{\left(\left| Z _1\right|^2+\left| Z _2\right|^2\right)}=5$