If `a, b, c` are in `GP`, then the equations `ax^2 +2bx+c = 0` and `dx^2 +2ex+f =0` have a common root if `d/a , e/b , f/c `are in

Explanation

Given equations are
`ax^(2)+2bx =c=0`……………..i
and `dx^(2)+2ex+f=0`……..ii
Since a,b,c are in GP.
`:.b^(2)=ac` or `b=sqrt(ac)`
From Eq. (i) `ax^(2)+2sqrt(ac)x+c=0`
or `(sqrt(ax)+sqrt(c))^(2)=0` ro `x=-(sqrt(c))/(sqrt(a))`
`:'` Given Eqs. (i) and (ii) have a common root.
Hence `x=-(sqrt(c))/(sqrt(a))` also satisfied Eq. (ii) then
`d(C/a)-2e(sqrt(c))/(sqrt(a))+f=0`
`impliesd/a)=(2e)/(sqrt(ac))+f/c=0`
or `d/a-(2e)/b+f/c=0 [ :' b=sqrt(ac)]`
or `d/a+f/c=(2e)/b`
`:.d/a,e/b,f/c` are in AP.
Hence `a/d,b/e,c/f` are in HP.