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

Question

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

If a,b, c are in G.P., then the equations `ax^(2) + 2bx + c = 0 and dx^(2) + 2ex + f = 0` have common root if `(d)/(a), (e)/(b), (f)/(c)` are in
A. A.P.
B. G.P.
C. H.P.
D. None of these

Monjur Alahi

4 views

2025-01-04 04:42

1 Answers

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Salim Ahmed Kawsar · Accepted Answer · 2023-02-05T15:29:48+06:00

Correct Answer - 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.