`a ,b ,c` are positive real numbers forming a G.P. ILf `a x 62+2b x+c=0a n ddx^2+2e x+f=0` have a common root, then prove that `d//a ,e//b ,f//c` are in A.P.

Options

A. `A.P.`
B. `G.P.`
C. `H.P.`
D. None of these

Answer

C

Explanation

`(c )` Let `alpha` be the root. Then `(a alpha^(2)+2blpha+c)+i(dalpha^(2)+2ealpha+f)=0`
`implies a alpha^(2)+2balpha+c=0impliesalpha=-sqrt((c )/(a))` (as `a`, `b`, `c` are in `G.P.`)
`:.` From `dalpha^(2)+2ealpha+f=0`
`d.(c )/(a)-2e(sqrt(c ))/(sqrt(a))+f=0`
`implies (d)/(a)-2.(e)/(b)+(f)/(c )=0`