First slide

Theory of equations

Question

If $α, β$ are the nonzero roots of ${ax}^{2} + bx + c = 0$ and $α^{2}, β^{2}$ are the roots of $a^{2} x^{2} + b^{2} x + c^{2} = 0$ , then a, b, c are in

Moderate

A

G.P
B

H.P
C

A.P
D

A.G.P

Solution

$α, β$ are roots of ${ax}^{2} + bx + c = 0$ . Hence,
$\begin{array}{l} α + β = - \frac{b}{a} (1) \\ αβ = \frac{c}{a} (2) \end{array}$
$α^{2}, β^{2}$ are roots of $a^{2} x^{2} + b^{2} x + c^{2} = 0$ . Hence,
$\begin{array}{l} α^{2} + β^{2} = - \frac{b^{2}}{a^{2}} (3) \\ α^{2} β^{2} = \frac{c^{2}}{a^{2}} (4) \end{array}$
Now, from (3), we have
$(α + β)^{2} - 2 αβ = - \frac{b^{2}}{a^{2}}$
$\Rightarrow {(\frac{- b}{a})}^{2} - 2 \frac{c}{a} = \frac{- b^{2}}{a^{2}}$
or $2 \frac{b^{2}}{a^{2}} = \frac{2 c}{a}$
or $b^{2} = ac \Rightarrow a, b, c$ are in G.P.

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