Suppose `alpha, beta` are roots of `ax^(2)+bx+c=0` and `gamma, delta` are roots of `Ax^(2)+Bx+C=0`. If `alpha,beta,gamma,delta` are in GP, then common

Question

Suppose `alpha, beta` are roots of `ax^(2)+bx+c=0` and `gamma, delta` are roots of `Ax^(2)+Bx+C=0`. If `alpha,beta,gamma,delta` are in GP, then common

Suppose `alpha, beta` are roots of `ax^(2)+bx+c=0` and `gamma, delta` are roots of `Ax^(2)+Bx+C=0`.
If `alpha,beta,gamma,delta` are in GP, then common ratio of GP is
A. `sqrt(((bA)/(aB)))`
B. `sqrt(((aB)/(bA)))`
C. `sqrt(((bC)/(cB)))`
D. `sqrt(((cB)/(bC)))`

Monjur Alahi

5 views

2025-01-04 04:42

1 Answers

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Salim Ahmed Kawsar · Accepted Answer · 2023-02-05T15:29:48+06:00

Correct Answer - B
`:.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.
`r=(beta)/(alpha)=(gamma)/(beta)=(delta)/(gamma)`
`implies alpha+beta=alpha+alphar=-(b)/(a)`
`implies alpha(1+r)=-(b)/(a)" " ".......(i)"`
and `gamma + delta=alphar^(2)+alphar^(3)=-(B)/(A)`
`implies alphar^(2)(1+r)=-(B)/(A)" " ".......(ii)"`
From Eqs. (i) and (ii), we get
`r^(2)=(Ba)/(bA)`
`r=sqrt((aB)/(bA))`.