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If a, b, c are in GP, then the equations...

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

A

AP

B

GP

C

HP

D

None of these

Text Solution

Verified by Experts

The correct Answer is:
A
Since, a,b, c are in GP
`rArr b^(2) = ac`
Given, `ax^(2) + 2bx + c = 0`
`rArr ax^(2) + 2sqrt(ac)x + c = 0` ltrbgt `rArr (sqrta x + sqrtc)^(2) = 0 rArr x = - sqrt((c)/(a))`
Since, `ax^(2) + 2bx + c = 0 and dx^(2) + 2ex + f = 0` have common root.
`:. x = - sqrt(c//a)` must satisfy
`dx^(2) + 2ex + f = 0`
`rArr d.(c)/(a) -2e sqrt((c)/(a)) + f = 0 rArr (d)/(a) - (2e)/(sqrt(ac)) + (f)/(c) = 0`
`rArr (2e)/(b) = (d)/(a) + (f)/(e) " " [ :' b^(2) = ac]`
Hence, `(d)/(a), (e)/(b), (f)/(c)` are in an AP
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