The origin and the roots of the equation `z^2 + pz + q = 0` form an equilateral triangle If -

Options

A. `p^(2)=q`
B. `p^(2)=3q`
C. `q^(2)=3p`
D. `q^(2)=p`

Answer

B

Explanation

Let `z_(1),z_(2)` be the roots of `z^(2)+pz+q=0` Then,
`z_(1)+z_(2)=-p,z_(1)z_(2)=q`
If `z_(1),z_(2),z_(3)` from an equilateral triangle, then
`z_(1)^(2)+z_(2)^(2)+z_(3)^(2)=z_(1)z_(2)+z_(2)z_(3)+z_(3)z_(1)`
`rArr z_(1)^(2)+z_(2)^(2)=z_(1)z_(2)` `[therefore z_(3)=0]`
`rArr (z_(1)+z_(2))^(2)=3z_(1)z_(2) rArr (-p)^(2)=3q rArr p^(2)=3q`