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If a,b,c are in GP, show that the equati...

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

A

A.P.

B

G.P.

C

H.P.

D

None of these

Text Solution

Verified by Experts

The correct Answer is:
A
a, b, c are in G.P `rArr b^(2) = ac`
Now the equation `ax^(2) + abx + c = 0` can be rewritten as `ax^(2) + 2sqrt(acx) + c = 0`
`rArr (sqrt(ax) + sqrt(c))^(2) = 0 rArr x = - sqrt((c)/(a)), - sqrt((c)/(a))`
If the two given equations have a common root, then this root must be `- sqrt((c)/(a))`
Thus `d (c)/(a) - 2e sqrt((c)/(a)) + f = 0 rArr (d)/(a) + (f)/(c) = (2e)/(c) sqrt((c)/(a)) = (2e)/(sqrt(ac)) = (2e)/(b) rArr (d)/(a), (e)/(b), (f)/(c)` are in A.P.
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