The roots of the cubic equation `(z+ab)^3=a^3,a !=0` represents the vertices of an equilateral triangle of sides of length

Options

A. `(1)/(sqrt(3))|ab|`
B. `sqrt(3)|a|`
C. `sqrt(3)|b|`
D. `|a|`

Answer

B

Explanation

Taking cub roots of both sides, we get
`z+ab=a(1)^(1//3) =a,aomega,aomega^(2)`
where `omega=-(1)/(2)+i(sqrt3)/(2),omega^(2)=-(1)/(2)-i(sqrt3)/(2)`
`therefore z_(1) =a-ab,z_(2)=aomega-ab,z_(3)=aomega^(2)-ab`
`|z_(1)-z_(2)|=|a(1-omega)|`
`=|a||1-(-(1)/(2)+i(sqrt3)/(2))|`
`=|a||(3)/(2)-i(sqrt(3))/(2)|`
`=|a|((9)/(4)+(3)/(4))^(1//2)=sqrt(3)|a|`
Similarly, `|z_(2)-z_(3)|=|z_(3)-z_(1)|=sqrt3|a|`