1.

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

Answer»


Solution : `|z_(1)| = 2`
Thisimplies that `z_(1)` lies on the circlehaving 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 thestraightline ` + y = 4sqrt(2)`

`|z_(1) -z_(2)|_("min")=` shortest distance betweencircleand straightline
= AB
`= OB - OA`
`= 4- 2=2`


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