Let ` alpha ` be the non-real 5 th root of unity. If ` z_1 and z_2 ` are two complex numbers lying on `|z| = 2`, then the value of ` sum_(t=0) ^(4) |z_1 + alpha ^(t)z_2 |^(2) ` is ______ .

Answer

40

Explanation

`|z_(1) +alpha^(t) z_(2)|^(2) = |z_(1)|^(2)+|z_(2)|^(2) + alpha^(t)z_(2)barz_(1) +bar((alpha))^(t) barz_(2)z_(1)`
Now `sum_(t=0)^(4) alpha^(t) =0 and sum_(t=0)^(4) bar((alpha))^(t) =0`
`therefore sum_(t=0)^(4) |z_(1)alpha^(t)z_(2)|^(2) = 5(|z_(1)|^(2) +|z_(2)|^(2))`
`= 5(4+4) = 40`