Let `alpha` and `beta` be roots of the equation `X^(2)-2x+A=0` and let `gamma` and `delta` be the roots of the equation `X^(2)-18x+B=0`. If `alpha lt beta lt gamma lt delta` are in arithmetic progression then find the valus of A and B.

Explanation

`therefore alpha,beta ,gamma, delta` are in AP.
Let`beta= alpha+d,gamma = alpha +2d,delta=alpha+3d,dgt0`
`" " " " " " ["here, sum of " alpha,beta,gamma,delta," is not given"]`
Given, `alpha + beta =2, alphabeta=A`
`implies 2alpha + d =2, alphabeta=A " " ".....(i)"`
and `gamma +delta=18, gammadelta=B`
` implies 2alpha+5d=18, gammadelta=B " " "......(ii)"`
From Eqs. (i) and (ii), we get
`d=4, alpha =-1`
`therefore beta=3, gamma=7, delta=11`
`implies A=alphabeta=(-1)(3)=-3`
and `B=gammadelta =(7)(11)=77`