Suppose `alpha, beta` are roots of `ax^(2)+bx+c=0` and `gamma, delta` are roots of `Ax^(2)+Bx+C=0`. If `a,b,c` are in GP as well as `alpha,beta, gamma, delta`, then `A,B,C` are in:

Options

A. (a) AP only
B. (b) GP only
C. (c) AP and GP
D. (d) None of these

Answer

B

Explanation

`:.alpha +beta=-(b)/(a),alphabeta=(c)/(a),alpha-beta=(sqrt(b^(2)-4ac))/(a)`
and `gamma+delta=-(B)/(A),gammadelta=(C)/(A),gamma-delta=(sqrt(B^(2)-4AC))/(A)`
Since, `alpha,beta, gamma,delta"......"` are in GP.
`:.(alpha)/(beta)=(gamma)/(beta)=(delta)/(gamma)`
`implies (beta)/(alpha)=(delta)/(gamma) implies (alpha)/(gamma)=(beta)/(delta)`
`implies (alpha+beta)/(gamma+delta)=sqrt((alphabeta)/(gammadelta))`
`implies(-(b)/(a))/(-(B)/(A))=sqrt(((c)/(a))/((C)/(A)))" " implies(b^(2)A^(2))/(a^(2)B^(2))=(cA)/(aC)`
`implies(acA^(2))/(aB^(2))=(cA)/(C)implies B^(2)=AC`
Hence, A,B,C are in GP.