Let `alpha,beta,gamma` are the real roots of the equation `x^3+a x^2+b x+c=0(a ,b ,c in Ra n da!=0)dot` If the system of equations `(inu ,v ,a n d w)` given by `alphau+betav+gammaw=0` `betau+gammav+alphaw=0` `gammau+alphav+betaw=0` has non-trivial solutions then the value of `a^2//b` is ________.

Answer

3

Explanation

Equation `x^(3) +ax^(2)+bx +c=0` has roots `alpha , beta , gamma` Therefore
`alpha+beta +gamma =-a`
`alpha beta + beta gamma +gamma alpha =b`
Since the given system of equation has non- trivial solutions we have
` |{:(alpha,,beta,,gamma),(beta,,gamma,,alpha),(gamma,,alpha,,beta):}|=0`
` " or " alpha^(3) +beta^(3)+gamma^(3) -3alpha beta gamma =0`
`" or " (alpha+beta +gamma) [alpha^(2) +beta^(2)+gamma^(2) -alphabeta -beta gamma - gamma alpha]=0`
`" or " (alpha+beta +gamma) [(alpha+beta+ gamma)^(2) -3 (alpha beta +beta gamma +gamma alpha)] =0`
`rArr -a[a^(2) -3b ]=0`
`" or " a^(2)//b=3`