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 AP, then common difference of AP is

Options

A. `(1)/(4)((b)/(a)-(B)/(A))`
B. `(1)/(3)((b)/(a)-(B)/(A))`
C. `(1)/(2)((c)/(a)-(B)/(A))`
D. `(1)/(3)((c)/(a)-(C)/(A))`

Answer

A

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` are in AP.
Let `beta=alpha+D,gamma=alpha+2D" and "delta=alpha+3D`
`:.alpha+ beta=(-b)/(a)" " implies alpha+alpha+D=-(b)/(a)`
or `2alpha +D=-(b)/(a)" " ".......(i)"`
and `gamma +delta=-(B)/(A)" " implies 2alpha +5D=-(B)/(A)" " "...........(ii)"`
From Eqs. (i) and(ii), we get
`4D=(-(B)/(A)+(b)/(a)) " or " D=(1)/(4)((b)/(a)-(B)/(A))`.