If x1 and x2 are the arithmetic and harmonic means of the roots of the equation ax2+bx+c=0, the equation whose quadratic roots are x1 and  x2, is

# If x1 and x2 are the arithmetic and harmonic means of the roots of the equation ${\mathrm{ax}}^{2}+\mathrm{bx}+\mathrm{c}=0$, the equation whose quadratic roots are x1 and  x2, is

1. A

${\mathrm{abx}}^{2}+\left({\mathrm{b}}^{2}+\mathrm{ac}\right)\mathrm{x}+\mathrm{bc}=0$

2. B

$2{\mathrm{abx}}^{2}+\left({\mathrm{b}}^{2}+4\mathrm{ac}\right)\mathrm{x}+2\mathrm{bc}=0$

3. C

$2{\mathrm{abx}}^{2}+\left({\mathrm{b}}^{2}+\mathrm{ac}\right)\mathrm{x}+\mathrm{bc}=0$

4. D

None of the above

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### Solution:

Let $\alpha$ and $\mathrm{\beta }$ be the roots of  ${\mathrm{ax}}^{2}+\mathrm{bx}+\mathrm{c}=0$

The required equation is

$\begin{array}{r}{\mathrm{x}}^{2}-\left[\left(-\frac{\mathrm{b}}{2\mathrm{a}}\right)+\left(-\frac{2\mathrm{c}}{\mathrm{b}}\right)\right]\mathrm{x}+\frac{2\mathrm{bc}}{2\mathrm{ab}}=0\\ 2{\mathrm{abx}}^{2}+\left({\mathrm{b}}^{2}+4\mathrm{ac}\right)\mathrm{x}+2\mathrm{bc}=0\end{array}$

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