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a,b,c are positive real numbers forming...

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

A

`A.P.`

B

`G.P.`

C

`H.P.`

D

None of these

Text Solution

Verified by Experts

The correct Answer is:
A
`(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`
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