If ` |z _1 |= 2 and (1 - i)z_2 + (1+i)barz_2 = 8sqrt2`, then the minimum value of ` |z_1 - z_2|` is ______.

Answer

2

Explanation

`|z_(1)| = 2`
This implies that `z_(1)` lies on the circle having centre at origin and radius 2.
`(1-i) z_(2) +(1+i) barz_(2) = 8sqrt(2)`
`therefore (1-i)(x_(2) + iy_(2)) +(1+i)(x_(2) -iy_(2)) = 8sqrt(2)`
`rArr x_(2) +y_(2) = 4sqrt(2)`
So, `z_(2)` lies on the straight line ` + y = 4sqrt(2)`

`|z_(1) -z_(2)|_("min")=` shortest distance between circle and straight line
= AB
`= OB - OA`
`= 4- 2=2`