`a ,b ,c` are positive real numbers forming a G.P. If `ax^2+2b x+c=0 and dx^2+2e x+f=0` have a common root, then prove that `d//a ,e//b ,f//c` are in A.P.

Explanation

For first equation `D = 4b^(2) - 4c = 0 ` (as given a, b, c are in G.P.) .The equation has equal roots which are equal to - `b//a` each. Thus, it should also be the root of the secound equation . Hence,
`d((-b)/(a))^(2) + 2e((-b)/(a)) + f = 0`
or `d(b^(2))/(a^(2))- 2(be)/(a) + f = 0`
or `d(ac)/(a^(2))- 2(be)/(a) + f = 0 " " (becauseb^(2) = ac)`
`(d)/(a)+ (f)/(c) = 2(be)/(ac) =2 (e)/(b)`.