First slide

Theory of equations

Question

The quadratic equation whose roots are A.M and H.M. between the roots of the equation $a x^{2} + b x + c = 0$ , is

Moderate

A

$a b x^{2} + (b^{2} + a c) x + b c = 0$
B

$2 a b x^{2} + (b^{2} + 4 a c) x + 2 b c = 0$
C

$2 a b x^{2} + (b^{2} + a c) x + b c = 0$
D

none of these

Solution

Let $α, β$ be the roots of the given equation. Then $α + β = - b / a and α β = c / a$ The A.M. and H.M. of a and $β$ are
$A. M = \frac{α + β}{2} and H . M . = \frac{2 α β}{α + β}$ respectively.
So, required equation is
$\begin{array}{l} x^{2} - x (\frac{α + β}{2} + \frac{2 α β}{α + β}) + (\frac{α + β}{2}) (\frac{2 α β}{α + β}) = 0 \\ 2 a b x^{2} + (b^{2} + 4 a c) x + 2 b c = 0 \end{array}$

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