If a, b and c are distinct integers and `omega``omega(ne1)` is a cube root of unity, then the minimum value of `|a+bomega+comega^(2)|+|a+bomega^(2)+comega|,` is

Options

A. `2sqrt(3)`
B. 3
C. `4sqrt(2)`
D. 2

Answer

A

Explanation

Let `z=a+bomega+comega^(2)`. Then,
`|z|^(2)=zbarz=(a+bomega+comega^(2))(a+bomega^(2)+comega) [therefore baromega=omega^(2)]`
`rArr |z|^(2)=1/2[(a-b)^(2)+(b-c)^(2)+(c-a)^(2)]`
`rArr |z|^(2)=1/2[(a-b)^(2)+(b-c)^(2)+(c-a)^(2)]`
`rArr |z|^(2) ge 1/2 xx 6 =3` `[therefore a ne b ne c therefore |a-b| gt 1, |b-c| ge 1` and `|a-c| ge 2]`
`rArr |z| ge sqrt(3)`
`therefore |a+bomega+comega^(2)|+|a+bomega^(2)+comega|`
`=|a+bomega+comegak^(2)|+|bar(a+bomega+comega^(2))|`
`=2|a+bomega+comega^(2)|=2|z| ge 2sqrt(3)`