The roots of the equation `t^3+3a t^2+3b t+c=0a r ez_1, z_2, z_3` which represent the vertices of an equilateral triangle. Then `a^2=3b` b. `b^2=a` c. `a^2=b` d. `b^2=3a`

Options

A. `a^(2) = 3b`
B. `b^(2) = a`
C. `a^(2) = a`
D. `b^(2) = 3a`

Answer

C

Explanation

`S_(1)=Sigmaz_(1)=-3a,S_(2)=Sigmaz_(1)z_(2)=3b`
`Sigmaz_(1)^(2)=Sigmaz_(1)z_(2)`
`rArr (Sigmaz_(1))^(2)-2Sigmaz_(1)z_(2)=Sigmaz_(1)z_(2)`
`rArr (Sigmaz_(1))^(2)=3Sigmaz_(1)z_(2)`
`rArr (-3a)^(2)=3(3b)`
`rArr a^(2)=b`