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If x(1) and x(2) are the arithmetic and ...

If `x_(1)` and `x_(2)` are the arithmetic and harmonic means of the roots of the equation `ax^(2)+bx+c=0`, the quadratic equation whose roots are `x_(1)` and `x_(2)` is

A

`abx^(2)+(b^(2)+ac)x+bc=0`

B

`2abx^(2)+(b^(2)+4ac)x+2bc=0`

C

`2abx^(2)+(b^(2)+ac)x+bc=0`

D

none of these

Text Solution

Verified by Experts

The correct Answer is:
B
Let `alpha` and beta` be the roots of `ax^(2)+bx+c=0`
`:.x_(1)=(alpha+beta)/2=-b/(2a)`
and `x_(2)=(2alpha beta)/(alpha+betsa)=(2 . c/a)/(-b/a)=-(2c)/b`
`:.` The required equation is
`x^(2)-[(-b/(2a))+(-(2c)/b)]x+(2bc)/(2ab)=0`
i.e. `2abx^(2)+(b^(2)+4ac)x+2bc=0`
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