Let `alpha ,beta` and `gamma` be the roots of the equations `x^(3) +ax^(2)+bx+ c=0,(a ne 0)`. If the system of equations `alphax + betay+gammaz=0`
`beta x +gamma y+ alphaz =0` and `gamma x =alpha y + betaz =0` has non-trivial solution then

Options

A. `a^(2)=3b`
B. `a^(3) =27c`
C. `b^(3) =27c^(2)`
D. `alpha+beta+gamma=0`

Answer

A, B, C

Explanation

Given system of equation has non-trivial solutions
`:. |{:(alpha,,beta,,gamma),(beta,,gamma,,alpha),(gamma,,alpha,,beta):}|=0`
`rArr alpha^(3) +beta^(3)+gamma^(3)=3alphabetagamma`
`rArr alpha=beta=gamma`
Now `alpha+beta +gamma=-a,alpha beta+ beta gamma +alpha gamma =b, alpha beta gamma =-c`
`rArr 3alpha =-a, 3alpha^(2) =b,alpha^(3)=-c`
`rArr alpha=-(a)/(3),alpha^(2)=(b)/(3),alpha^(3)=-c`
`rArr a^(2) =3b,a^(3) =27 c, b^(3) =27 c^(2)`